Bryna Kra
email: kra@math.northwestern.edu

Furstenberg Weiss Bergelson

NIL-BOHR SETS OF INTEGERS

We study relations between subsets of integers that are large, where large can be interpreted in terms of size (such as a set of positive upper density or a set with bounded gaps) or in terms of additive structure (such as a Bohr set). Bohr sets are fundamentally abelian in nature and are linked to Fourier analysis.

Recently it has become apparent that a higher order, non-abelian, Fourier analysis plays a role in both additive combinatorics and in ergodic theory. Here we introduce a higher order version of Bohr sets and give various properties of these objects, generalizing results of

1. Introduction 1.1. Additive combinatorics and Bohr sets. Additive combinatorics is the study of structured subsets of integers, concerned with questions such as what one can say about sets of integers that are large in terms of size or about sets that are large in terms of additive structure. An interesting problem is finding various relations between classes of large sets.

Sets with positive upper Banach density or syndetic sets 1 are examples of sets that are large in terms of size. A simple result relates these two notions: if A ⊂ Z has positive upper Banach density, then the set of differences ∆(A) = A -A = {ab : a, b ∈ A} is syndetic.

An example of a structured set is a Bohr set. Following a modification of the traditional definition introduced in [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF], we say that a subset A ⊆ Z is a Bohr set if there exist m ∈ N, α ∈ T m , and an open set

The first author was partially supported by the Institut Universitaire de France and the second by NSF grant 0555250 and by the Clay Mathematics Institute. This work was begun during the visit of the authors to MSRI and completed while the second author was a visitor at Institut Henri Poincaré; we thank the institutes for their hospitality. The set A ⊂ Z is said to be syndetic if it intersects every sufficiently large interval.

1 U ⊂ T m such that {n ∈ Z : nα ∈ U} is contained in A (see Definition 2.1). It is easy to check that the class of Bohr sets is closed under translations. Most of the notions of a large set that are defined solely in terms of size are also closed under translation. However, we have important classes of structure sets that are not closed under translation. One particular example is that of a Bohr 0 -set: a subset A ⊆ Z is a Bohr 0set if it is a Bohr set such that the set U in the previous definition contains 0.

A simple application of the pigeonhole principle gives that if S is an infinite set of integers, then S -S has nontrivial intersection with every Bohr 0 -set. This is another example of largeness: a set is large if it has nontrivial intersection with every member of some class of sets. Such notions of largeness are generally referred to as dual notions and are denoted with a star. For example, a ∆ * -set is a set that has nontrivial intersection with the set of differences ∆(A) from any infinite set A.

Here we study converse results. If a set intersects every set of a given class, then our goal is to show that it has some sort of structure. Such theorems are not, in general, exact converses of the direct structural statements. For example, there exist ∆ * -sets that are not Bohr 0 -sets (see [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF]). But, this statement is not far from being true. Strengthening a result of [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF], we show (Theorem 2.8) that a ∆ * -set is a piecewise Bohr 0 -set, meaning that it agrees with a Bohr 0 -set on a sequence of intervals whose lengths tend to infinity.

1.2. Nil-Bohr sets. Bohr sets are fundamentally linked to abelian groups and Fourier analysis. In the past few years, it has become apparent in both ergodic theory and additive combinatorics that nilpotent groups and a higher order Fourier analysis play a role (see, for example, [START_REF] Gowers | A new proof of Szemerédi's Theorem[END_REF], [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF], and [START_REF] Green | Linear equations in the primes[END_REF]). As such, we define a d-step nil-Bohr 0 -set, analogous to the definition of a Bohr 0 -set, but with a nilmanifold replacing the role of an abelian group (see Definition 2.3). For d = 1, the abelian case, this is exactly the object studied in [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF]. Here we generalize their results for d ≥ 1.

We obtain a generalization of Theorem 2.8 on different sets, introducing the idea of a set of sums with gaps. For an integer d ≥ 0 and an infinite sequence P = (p i ) in N, the set of sums with gaps of length < d of P is defined to be the set SG d (P ) of all integers of the form

ǫ 1 p 1 + ǫ 2 p 2 + • • • + ǫ n p n ,
where n ≥ 1 is an integer, ǫ i ∈ {0, 1} for 1 ≤ i ≤ n, the ǫ i are not all equal to 0, and the blocks of Our main result (Theorem 2.6) is that a set with nontrivial intersection with any SG d -set is a piecewise d-step nilpotent Bohr 0 -set.

1.3. The method. The first ingredient in the proof is a modification and extension of the Furstenberg Correspondence Principle. The classical Correspondence Principle gives a relation between sets of integers and measure preserving systems, relating the size of the sets of integers to the measure of some sets of the system. It does not give relations between structures in the set of integers under consideration and ergodic properties of the corresponding system. Some information of this type is provided by our modification (originally introduced in [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF]).

We then are left with studying certain properties of the systems that arise from this correspondence. As in several related problems, the properties of the system that we need are linked to certain factors of the system, which are nilsystems. This method and these factors were introduced in the study of convergence of some multiple ergodic averages in [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF].

Working within these factor systems, we conclude by making use of techniques for the analysis of nilsystems that have been developed over the last few years. In the abelian setting, a fundamental tool is the Fourier transform, but no analog exists for higher order nilsystems2 . Another classical tool available in the abelian case is the convolution product, but this too is not defined for general nilsystems. Instead, in Section 4 we build some spaces and measures that take on the role of the convolution. As an example, if G is a compact abelian group we can consider the subgroup

{(g 1 , g 2 , g 3 , g 4 ) ∈ G 4 : g 1 + g 2 = g 3 + g 4 }
of G 4 , and we take integrals with respect to its Haar measure. This replaces the role of the convolution product.

These constructions are then used to prove the key convergence result (Proposition 5.4). By studying the limit, Theorem 2.6 is deduced in Section 6. By further iterations, Theorems 2.8 and 2.10 are proved in Section 7.

Precise statements of definitions and results

2.1. Bohr sets and Nil-Bohr sets. We formally define the objects described in the introduction:

Definition 2.1. A subset A ⊆ Z is a Bohr set if there exist m ∈ N, α ∈ T m , and an open set U ⊂ T m such that {n ∈ Z : nα ∈ U} is contained in A; the set A is a Bohr 0 -set if additionally 0 ∈ U.
Note that these sets can also be defined in terms of the topology induced on Z by embedding the integers into the Bohr compactification: a subset of Z is Bohr if it contains an open set in the induced topology and is Bohr 0 if it contains an open neighborhood of 0 in the induced topology.

We can generalize the definition of a Bohr 0 -set for return times in a nilsystem, rather than just in a torus. We first give a short definition of a nilsystem and refer to Section 3.2 for further properties. Definition 2.2. If G is a d-step nilpotent Lie group and Γ ⊂ G is a discrete and cocompact subgroup, the compact manifold X = G/Γ is a d-step nilmanifold. The Haar measure µ of X is the unique probability measure that is invariant under the action x → g • x of G on X by left translations.

If T denotes left translation on X by a fixed element of G, then (X, µ, T ) is a d-step nilsystem.

Using neighborhoods of a point, we define a generalization of a Bohr set:

Definition 2.3. A subset A ⊆ Z is a Nil d Bohr 0 -set if there exist a d-step nilsystem (X, µ, T ), x 0 ∈ X, and an open set U ⊂ X containing x 0 such that {n ∈ Z : T n x 0 ∈ U} is contained in A.
Similar to the Bohr compactification of Z that can be used to define the Bohr sets, there is a d-step nilpotent compactification of Z that can be used to define the Nil d Bohr 0 -sets. This compactification is a non-metric compact space Z, endowed with a homeomorphism T and a particular point x 0 with dense orbit, and is characterized by the following properties: i) Given any d-step nilsystem (Z, T ) and a point x 0 ∈ Z, there is a unique factor map π Z :

Z → Z with π Z ( x 0 ) = x 0 .
ii) The topology of Z is spanned by these factor maps π Z .

Remark. A Bohr 0 -set can be defined in terms of almost periodic sequences. In the same way, a Nil d Bohr 0 -set can be defined in terms of some particular sequences, the d-step nilsequences. Since Nil d Bohr 0sets are defined locally, it seems likely that they can be defined by certain particular types of nilsequences, namely those arising from generalized polynomials without constant terms. We do not address this issue here.

Piecewise versions.

If F denotes a class of subsets of integers, various authors, for example Furstenberg in [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF] and Bergelson, Furstenberg, and Weiss in [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF], define a subset A of integers to be a piecewise-F set if A contains the intersection of a sequence of arbitrarily long intervals and a member of F . For example, the notions of piecewise-Bohr set, a piecewise-Bohr 0 -set, and a piecewise-Nil d Bohr 0 -set, can be defined in this way. However, the notion of a piecewise set is rather weak: for example, a piecewise-Bohr set defined in this manner is not necessarily syndetic. The properties that we can prove are stronger than the traditional piecewise statements, and in particular imply the traditional piecewise versions. For this, we introduce a stronger definition of piecewise: Definition 2.4. Given a class F of subsets of integers, the set A ⊂ Z is said to be strongly piecewise-F , written PW-F , if for every sequence (J k : k ≥ 1) of intervals whose lengths |J k | tend to ∞, there exists a sequence (I j : j ≥ 1) of intervals satisfying: i) For each j ≥ 1, there exists some k = k(j) such that the interval I j is contained in J k ; ii) The lengths |I j | tend to infinity; iii) There exists a set Λ ∈ F such that Λ ∩ I j ⊂ A for every j ≥ 1.

Note that Λ depends on the sequence (J k ). With this definition of strongly piecewise, if the class F consists of syndetic sets then every PW-F -set is syndetic. In particular, a strongly piecewise-Bohr set, denoted PW-Bohr, is syndetic. Similarly, we denote a strongly piecewise-Bohr 0 -set by PW-Bohr 0 and a strongly piecewise-Nil d Bohr 0 -set by PW-Nil d Bohr 0 and these sets are also syndetic.

Sumsets and Difference Sets.

Definition 2.5. Let E ⊂ N be a set of integers. The sumset of E is the set S(E) consisting of all nontrivial finite sums of distinct elements of E.

A subset A of N is a S * r -set if A ∩ S(E) = ∅ for every set E ⊂ N with |E| = r.
We have: Theorem 2.6. Every S * d+1 -set is a PW-Nil d Bohr 0 -set. We can iterate this result, leading to the following definitions from [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF] and [START_REF] Furstenberg | Recurrence in ergodic theory and combinatorial number theory[END_REF]:

Definition 2.7. If S is a nonempty subset of N, define the difference set ∆(S) by ∆(S) = (S -S) ∩ N = {b -a : a ∈ S, b ∈ S, b > a} . If A is a subset of N, A is a ∆ * r -set if A ∩ ∆(S) = ∅ for every subset S of N with |S| = r; A is a ∆ * -set if A ∩ ∆(S) = ∅ for every infinite subset S of N. Theorem 2.8. Every ∆ * -set is a PW-Bohr 0 -set. Every ∆ *
r set is obviously a ∆ * -set and Theorem 2.8 generalizes a result of [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF]. The class of sets of the form ∆(S) with |S| = 3 coincides with the class of sets of the form S(E) with |E| = 2 and thus the classes ∆ * 3 and S * 2 are the same. Theorem 2.8 generalizes the case d = 1 of Theorem 2.6.

The converse statement of Theorem 2.8 does not hold. However, it is easy to check that every Bohr 0 -set is a ∆ * -set (see [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF]). Definition 2.9. Let d ≥ 0 be an integer and let P = (p i ) be a (finite or infinite) sequence in N. The set of sums with gaps of length < d of P is the set SG d (P ) of all integers of the form

ǫ 1 p 1 + ǫ 2 p 2 + • • • + ǫ n p n ,
where n ≥ 1 is an integer, ǫ i ∈ {0, 1} for 1 ≤ i ≤ n, the ǫ i are not all equal to 0, and the blocks of consecutive 0's between two 1's have length < d.

A subset

A ⊆ N is an SG * d -set if A ∩ SG d (P ) = ∅ for every infinite sequence P in N.
Note that in this definition, P is a sequence and not a subset of N. For example, if P = {p 1 , p 2 , . . . }, then SG 1 (P ) is the set of all sums p m + • • • + p n of consecutive elements of P , and thus it coincides with the set ∆(S) where S = {s, s + p 1 , s + p 1 + p 2 , . . . }. Therefore SG * 1 -sets are the same as ∆ * -sets.

For a sequence P , SG 2 (P ) consists of all sums of the form

m 1 i=m 0 p i + m 2 i=m 1 +2 p i + • • • + m k i=m k-1 +2 p i + m k+1 i=m k +2 p i ,
where k ∈ N and m 0 , m 1 , . . . , m k+1 are positive integers satisfying m i+1 ≥ m i + 2 for i = 0, . . . , k. In general, a Nil d Bohr 0 -set is not a ∆ * -set. To construct an example, take an irrational α and let Ω be the set of n ∈ Z such that n 2 α is close to 0 mod 1. Then Ω is a Nil 2 Bohr 0 -set, as can be checked by considering the transformation on T 2 defined by (x, y) → (x+ α, y + x). On the other hand, by induction we can build an increasing sequence n j of integers such that n 2 j α is close to 1/3 mod 1, while n i n j α mod 1 is close to 0 for i < j. Taking S to be the set of such n j , we have that ∆(S) does not intersect Ω.

This leads to the following question:

Preliminaries

3.1. Notation. We introduce notation that we use throughout the remainder of the article. If X is a set and d ≥ 1 is an integer, we write X [d] = X 2 d and we index the 2 d copies of X by {0, 1} d . Elements of X [d] are written as

x = (x ǫ : ǫ ∈ {0, 1} d ) .
We write elements of {0, 1} d without commas or parentheses.

We also often identify {0, 1} d with the family P([d]) of subsets of [d] = {1, 2, . . . , d}. In this identification, ǫ i = 1 is the same as i ∈ ǫ and ∅ = 00 . . . 0.

For ǫ ∈ {0, 1} d and n ∈ Z d , we write |ǫ| = ǫ 1 + . . . + ǫ d and ǫ

• n = ǫ 1 n 1 + . . . + ǫ d n d .
If p : X → Y is a map, then we write p [d] : X [d] → Y [d] for the map (p, p, . . . , p) taken 2 d times. In particular, if T is a transformation on the space X, we define T [d] : X [d] → X [d] as T × T × . . . × T taken 2 d times. We define the face transformations T

[d] i for 1 ≤ i ≤ d by: (T [d] i x) ǫ = T (x ǫ ) if ǫ i = 1 x ǫ otherwise .
In a slight abuse of notation, we denote all transformations, even in different systems, by the letter T (unless the system is naturally a Cartesian product).

For convenience, we assume that all functions are real valued.

3.2. Review of nilsystems.

Definition 3.1. If G is a d-step nilpotent Lie group and Γ ⊂ G is a discrete and cocompact subgroup, the compact manifold X = G/Γ is a d-step nilmanifold.
The Haar measure µ of X is the unique probability measure that is invariant under the action x → g • x of G on X by left translations.

If T denotes left translation on X by a fixed element of G, then (X, µ, T ) is a d-step nilsystem.

(We generally omit the σ-algebra from the notation, writing (X, µ, T ) for a measure preserving system rather than (X, B, µ, T ), where B denotes the Borel σ-algebra.)

A d-step nilsystem is an example of a topological distal dynamical system. For a d-step nilsystem, the following properties are equivalent: transitivity, minimality, unique ergodicity, and ergodicity. (Note that the first three of these properties refer to the topological system, while the last refers to the measure preserving system.) Also, the closed orbit of a point in a d-step nilsystem is a d-step nilsystem, and it follows that this closed orbit is minimal and uniquely ergodic. See [START_REF] Auslander | Flows on homogeneous spaces[END_REF] for proofs and general references on nilsystems.

We also speak of a nilsystem (X = G/Γ, T 1 , . . . , T d ), where T 1 , . . . , T d are translations by commuting elements of G. All the above properties hold for such systems.

We also make use of inverse limits of systems, both in the topological and measure theoretic senses. All inverse limits are implicitly assumed to be taken along sequences. Inverse limits for a sequence of nilsystems are the same in both the topological and measure theoretic senses: this follows because a measure theoretic factor map between two nilsystems is necessarily continuous.

Many properties of the nilsystems also pass to the inverse limit. In particular, in an inverse limit of d-step nilsystems, every closed orbit is minimal and uniquely ergodic.

3.3. Structure Theorem. Assume now that (X, µ, T ) is an ergodic system.

We recall a construction and definitions from [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF], but for consistency we make some small changes in the notation. For an integer d ≥ 0, a measure µ [d] on X [d] was built in [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF]. Here we denote this measure by µ (d) .

The measure µ (d) is invariant under T [d] and under all the face transformations

T [d] i , 1 ≤ i ≤ d. Each of the projections of the measure µ (d) on X is equal to the measure µ.
If f is a bounded measurable function on X, then

ǫ⊂[d] f (x ǫ ) dµ (d) (x) ≥ 0
and we define |||f ||| d to be this expression raised to the power 1/2 d . Then ||| • ||| d is a seminorm on L ∞ (µ). A main result from [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF] is that this is a norm if and only if the system is an inverse limit of (d -1)-step nilsystems. More precisely, a summary of the Structure Theorem of [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF] is:

Theorem 3.2. Assume that (X, µ, T ) is an ergodic system. Then for each d ≥ 2, there exist a system (Z d , µ d , T ) and a factor map

π d : X → Z d satisfying: i) (Z d , µ d , T ) is the inverse limit of a sequence of (d -1)-step nil- systems. ii) For each f ∈ L ∞ (µ), |||f -E(f | Z d ) • π d ||| d = 0.
For each d ≥ 1, we call (Z d , µ d , T ) the HK-factor of order d of (X, µ, T ). The factor map π d : X → Z d is measurable, and a priori has no reason to be continuous. For ℓ ≤ d, Z ℓ is a factor of Z d , with a continuous factor map.

If (X, µ, T ) is an ergodic inverse limit of (d -1)-step nilsystems, we define3 X (d) to the closed orbit in X [d] of a point x 0 = (x 0 , . . . , x 0 ) (for some arbitrary x 0 ∈ X) under the transformations T [d] and d) , endowed with these transformations, is minimal and uniquely ergodic. Its unique invariant measure is exactly the measure µ (d) described above. When (X, µ, T ) is a (d -1)-step nilsystem, then X (d) is a nilmanifold and µ (d) is its Haar measure.

T [d] i for 1 ≤ i ≤ d. The system X (
3.4. Furstenberg correspondence principle revisited. By ℓ ∞ (Z), we mean the algebra of bounded real valued sequences indexed by Z.

Let A be a subalgebra of ℓ ∞ (Z), containing the constants, invariant under the shift, closed and separable with respect to the norm • ∞ of uniform convergence. We refer to this simple as "an algebra." In applications, finitely many subsets of Z are given and A is the shift invariant algebra spanned by indicator functions of these subsets.

Given an algebra, we associate various objects to it: a dynamical system, an ergodic measure on this system, a sequence of intervals, etc. We give a summary of these objects without proof, referring to [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF] for more information.

A system associated to

A. There exist a topological dynamical system (X, T ) and a point x 0 ∈ X such that the map

φ ∈ C(X) → φ(T n x 0 ) : n ∈ Z ∈ ℓ ∞ (Z)
is an isometric isomorphism of algebras from C(X) onto A. (We use C(X) to denote the collection of continuous functions on X.)

In particular, if S is a subset of Z with 1 S ∈ A, then there exists a subset S of X that is open and closed in X such that [START_REF] Auslander | Flows on homogeneous spaces[END_REF] for every n ∈ Z, T n x 0 ∈ S if and only if n ∈ S .

Some averages and some measures associated to A.

There also exist a sequence I = (I j : j ≥ 1) of intervals of Z, whose lengths tend to infinity, and an invariant ergodic probability measure µ on X such that

(2) for every φ ∈ C(X),

1 |I j | n∈I j φ(T n x 0 ) → φ dµ as j → +∞ .
Given a subset S of Z, we can chose the intervals I j such that

|S ∩ I j | |I j | → d * (S) as j → +∞ ,
where d * (S) denotes the upper Banach density of S.

In particular, we can assume that the intervals I j are contained in N. We omit the subscripts n and/or I if they are clear from the context.

3.4.4.

Averages and factors of order k. Recall that Z k denotes the HKfactor of order k of (X, µ, T ) and that π k : X → Z k denotes the factor map.

The sequence of intervals I = (I j : j ≥ 1) can be chosen such that:

Proposition 3.3. For every k ≥ 1, there exists a point e k ∈ Z k such that π ℓ,k (e k ) = e ℓ for ℓ < k and such that for every φ ∈ C(X) and every

f ∈ C(Z k ), lim Av I φ(T n x 0 )f (T n e k ) = φ • f • π k dµ = E(φ | Z k ) f dµ k .
This formula extends (2). The next corollary is an example of the relation between integrals on the factors Z k and PW-Nil Bohr-sets. More precise results are proved and used in the sequel. Corollary 3.4. Let S be a subset of Z such that 1 S belongs to the algebra A and let S be the corresponding subset of X. Let f be a nonnegative continuous function on Z k with f (e k ) > 0, where e k is as in Proposition 3.3. If

1 e S (x) • f • π k (x) dµ(x) = 0 then Z \ S is a PW-Nil k Bohr 0 -set. Proof. Let Λ = {n ∈ Z : f (T n e k ) > f (e k )/2}.
Then Λ is a Nil k Bohr 0set. By Proposition 3.3 and definition (1) of S, the averages on

I j of 1 S (n)f (T n e k ) converge to zero. Thus lim j→+∞ |I j ∩ S ∩ Λ| |I j | = 0 .
Therefore, the subset E = j I j \ (S ∩ Λ) contains arbitrarily long intervals J ℓ , ℓ ≥ 1. For every ℓ, J ℓ ∩ (Z \ S) ⊃ J ℓ ∩ Λ.

3.4.5. It is easy to check that given a sequence of intervals (J k : k ≥ 1) whose lengths tend to infinity, we can choose the intervals (I j : j ≥ 1) satisfying all of the above properties, and such that each interval I j is a subinterval of some J k . To see this, we first reduce to the case that the intervals J k are disjoint and separated by sufficiently large gaps.

We set S to be the union of these intervals. We have d * (S) = 1 and we can choose intervals

I ′ j with |S ∩ I ′ j |/|I ′ j | → 1. For every j ∈ N, there exists k j such that |I ′ j ∩ J k j |/|I ′ j | → 1 as j → +∞.
We set I j = I ′ j ∩ J k j and the sequence (I j : j ≥ 1) satisfies all the requested properties.

3.5. Definition of the uniformity seminorms. We recall definitions and results of [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF] adapted to the present context. We keep notation as in the previous sections; in particular, Z k and e k are as in Proposition 3.3.

Let I be as in Section 3.4 and let B be the algebra spanned by A and sequences of the form (f (T n e k ) : n ∈ Z), where f is a continuous function on Z k for some k. By Proposition 3.3, for every sequence a = (a n : n ∈ Z) belonging to the algebra B, the limit lim Av I,n a n exists.

Given a sequence a ∈ B,

for h = (h 1 , . . . , h d ) ∈ Z d , let c h = lim Av I,n ǫ⊂[d] a n+ǫ•h . Then lim H→∞ 1 H d H-1 h 1 ,...,h d =0 c h
exists and is nonnegative. We define a I,d to be this limit raised to the power 1/2 d . Proposition 3.5. Let (Z, T ) be an inverse limit of k-step nilsystems and f be a continuous function on Z. Then for every δ > 0 there exists C = C(δ) > 0 such that for every sequence a = (a n : n ∈ Z) belonging to the algebra B and for every z ∈ Z,

limsup Av I a n f (T n z) ≤ δ a ∞ + C a I,k+1 .
Proof. By density, we can reduce to the case that (Z, T ) is a k-step nilsystem and that the function f is smooth. In this case, the result is contained in [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF] under the hypothesis that the system is ergodic. Indeed, by Proposition 5.6 of this paper, f is a "dual function" on X. By the "Modified Direct Theorem" of Section 5.4 in [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF], there exists a constant

|||f ||| * k ≥ 0 with limsup Av I a n f (T n z) ≤ |||f ||| * k • a I,k+1 .
In the proofs of [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF] we can check that the hypothesis of ergodicity is not used.

The next proposition was proved in [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF] and follows from the Structure Theorem.

Proposition 3.6. Let φ be a continuous function on X with |φ| ≤ 1, k ≥ 1 an integer, and f a continuous function on

Z k with |f | ≤ 1. Then φ(T n x 0 ) -f (T n e k ) : n ∈ Z I,k+1 ≤ 2 f -E(φ | Z k ) 1/2 k+1 L 1 (µ k ) .
4. Some measures associated to inverse limits of nilsystems 4.1. Standing assumptions. We assume that every topological system (Z, T ) is implicitly endowed with a particular point, called the base point. Every topological factor map is implicitly assumed to map = base point to base point. For every k ≥ 1, we take the base point of Z k to be the point e k introduced in Section 3.4.4.

If (Z, T ) is a nilsystem with Z = G/Γ, then by changing the group Γ if needed, we can assume that the base point of Z is the image in Z of the unit element of G.

The measures µ (m)

e . Proposition 4.1. Let (X, µ, T ) be an ergodic inverse limit of ergodic k-step nilsystems, endowed with the base point e ∈ X, and let m ≥ 1 be an integer. a) The closed orbit of the point e [m] = (e, e, . . . , e) of X (m) under the transformations T

[m] i , 1 ≤ i ≤ m, is X (m) e = {x ∈ X (m) : x ∅ = e} . b) Let µ (m) e
be the unique measure on this set invariant under these transformations. Then the image of µ

(m) e under each of the natural projections x → x ǫ : X [m] → X, ∅ = ǫ ⊂ [d], is equal to µ. c) Let (Y,
ǫ , ∅ = ǫ ⊂ [d], are 2 m -1 bounded measurable functions on X, ∅ =ǫ⊂[d] f ǫ (x ǫ ) dµ (m) e (x) = ∅ =ǫ⊂[d] E(f ǫ | Y )(y ǫ ) dν (m) e (y)
.

(The existence of these integrals follows from b).)

Following our convention, we assume in c) and d) that Y is endowed with a base point and that p maps the base point to the base point.

Proof. We first prove a) and d) assuming that X is a nilsystem. (While the proof is contained in [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF], we sketch it here in order to introduce some objects and some notation.)

For g ∈ G and F ⊂ P([m]), we write g F for the element of G [m] given by

for every ǫ ⊂ [m], g F ) ǫ = g if ǫ ∈ F ; 1 otherwise.
We write X = G/Γ and let τ be the element of G defining the transformation T of X. We can assume that the base point e of X is the image in X of the unit element of G. Since (X, µ, T ) is ergodic, we can also assume that G is spanned by the connected component of the identity and τ . We recall a convenient presentation of G (m) (see [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF] and [START_REF] Green | Linear equations in the primes[END_REF]).

Let α 1 , . . . , α 2 m be an enumeration of all subsets of [d] such that |α i | is increasing. In particular, α 1 = ∅. For 1 ≤ i ≤ 2 m , let F i = {ǫ : α i ⊂ ǫ ⊂ [m]}. For every i, F i is an upper face of the cube P([m]), meaning a face containing the vertex [d]; its codimension is |α i |. Then F 1 , . . . , F 2 m is an enumeration of all the upper faces, in decreasing order of codimension. In particular, F 1 is the whole cube P([m]).

Each element of G (m) can be written in a unique way as

(3) h = g F 1 1 g F 2 2 . . . g F 2 d 2 d , where g i ∈ G |α i | for every i . (By convention, G 0 = G.)
We define

G (m) e = g ∈ G (m) : g ∅ = 1 .
This group is closed and normal in G (m) and every element of G (m) can be written in a unique way as

h [m] g with h ∈ G and g ∈ G (m) e . Moreover, G (m) e
is the set of elements of G (m) that are written as in (3) with g 1 = 1. From this, it is easy to deduce that the commutator subgroup of this group is equal to

G (m) e ∩ (G 2 ) [m] . Clearly, the subset X (m) e of X (m) is invariant under G (m) e
and it follows from the preceding description that the action of this group on this set is transitive. Therefore, the subgroup

Γ (m) e := Γ (m) ∩ G (m) e of G (m) e is cocompact in G (m) e
and we can identify X is invariant under translation by this element. The result follows.

We now turn to the proof of the proposition in the general case. For a), the generalization to inverse limits is immediate.

b) Let ǫ ∈ [d] with ǫ = ∅. Let i ∈ ǫ. Then for every x ∈ X (m) we have T x ǫ = (T [m] i x) ǫ . Since the measure µ (m) e is invariant under T (m) i
, its image under the projection x → x ǫ is invariant under T and thus is equal to µ.

Property c) is immediate. d) Let the functions f ǫ be as in the statement; without loss we can assume that |f ǫ | ≤ 1 for every ǫ.

Let (X i , µ i , T i ), i ≥ 1, be an increasing sequence of k-step nilsystems with inverse limit (X, µ, T ) and let π i : X → X i , i ≥ 1, be the (pointed) factor maps.

For every ǫ, ∅ = ǫ ⊂ [d], we have that

f ǫ -E(f ǫ • X i ) • π i L 1 (µ) → 0 as i → +∞
and thus (4)

∅ =ǫ⊂[d] E(f ǫ | X i ) • π i (x ǫ ) dµ (m) e (x) → ∅ =ǫ⊂[d] f ǫ (x ǫ ) dµ (m) e (x)
as i → +∞. For every i, let (W i , σ i , T ) be the (m -1)-step factor of X i , q i : X i → W i the factor map and r i = q i • π i .

We have showed above that, for every i, the measure µ i (m) e is relatively independent with respect to σ i (m) e . By using c) twice, we get that the second integral in ( 4) is equal to

∅ =ǫ⊂[d] E(f ǫ | W i ) • r i (x ǫ ) dµ (m) e (x) .
As the systems X i form an increasing sequence, the systems W i also form an increasing sequence. Let (W, σ, T ) be the inverse limit if this sequence. This system is a factor of X, and writing r : X → W for the factor map, we have that

E(f ǫ | W i ) • r i → E(f ǫ | W ) • r in L 1 (µ) for every ǫ. We get (5) ∅ =ǫ⊂[d] f ǫ (x ǫ ) dµ (m) e (x) = ∅ =ǫ⊂[d] E(f ǫ | W ) • r(x ǫ ) dµ (m) e (x) .
This means that the measure µ (m) e is relatively independent with respect to σ (m) e . As W is an inverse limit of (m -1)-step nilsystems and is a factor of X, it is a factor of the (m -1)-step factor Y if X. If for some ǫ we have E(f ǫ | Y ) = 0, then we have E(f ǫ | W ) = 0 and the second integral in ( 5) is equal to zero. The result follows.

Passing to inverse limits adds technical issues to each proof. These issues are not difficult and the passage to inverse limits uses only routine techniques, as in the preceding proof. However, it does greatly increase the length of the arguments, and so in general we omit this portion of the argument.

The measures µ

(m) e,x . In this section, again (X, µ, T ) is an ergodic inverse limit of k-step nilsystems, with base point e ∈ X.

For x ∈ X we write

X (m) e,x = {x ∈ X (m) : x ∅ = e and x {m} = x} . The set X (m)
e,e is the image of the set X (m,1) e introduced below by a permutation of coordinates.

x → x ǫ : X [m] → X, ǫ = ∅, ǫ = {m}, is equal to µ. ii) If f ǫ , ǫ ⊂ [m], ǫ ⊂ [1]
, are 2 m -2 bounded measurable functions on X, then the function F on X given by

F (x) = ǫ⊂[d] ǫ =∅, ǫ ={m} f ǫ (x ǫ ) dµ (m) e,x (x)
is continuous. iii) Moreover, for every bounded measurable function f on X,

f (x)F (x) dµ(x) = f (x {m} ) ǫ⊂[d] ǫ =∅,[m] f ǫ (x ǫ ) dµ (m)
e (x) .

Proof. It suffices to prove this Proposition in the case that (X, µ, T ) is k-step nilsystem, as the general case follows by standard methods.

We write X = G/Γ as usual. We can assume that e is the image in X of the unit element 1 of G. We define

G (m)
e,e = g ∈ G (m) : g ∅ = g {m} = 1 . This group is closed and normal in G. It is the set of elements of G (m) that can be written as in (3) with g ∅ = 1 and g i = 1 for the value of i such that α i = {m}. Recall that e [m] = (e, e, . . . , e).

It is easy to check that G e,e under g F and h F are the same.

Therefore, for every x ∈ X we can define a measure µ

(m) e,x on X (m)
e,x by ( 6)

µ (m) e,x = g F • µ (m)
e,e for every g ∈ G such that g • e = x .

In particular, for every h ∈ G and every x ∈ X,

(7) µ (m) e,h•x = h F • µ (m) e,x .
If T is the translation by τ ∈ G, then T

[m] m is the translation by τ F and and so for every integer n, [START_REF] Leibman | Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold[END_REF] µ

(m) e,T n x = T [m] m n • µ (m) e,x .
For

1 ≤ i < m, τ [m] i ∈ G (m)
e,e and thus, for every x ∈ X, µ

(m)
e,x is invariant under T

[m] i . As above, it follows that this measure satisfies the first property of the proposition.

To prove the other properties, the first statement of the proposition implies that we can reduce to the case that the functions f ǫ are continuous. By ( 6), the map x → µ (m) e,x is weakly continuous and the function F is continuous. We are left with showing that

µ (m) e = µ (m)
e,x dµ(x) .

For 1 ≤ i < m, since for every x the measure µ

(m) e,x is invariant under T [m]
i , the measure defined by this integral is invariant under this transformation. By [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF], µ

(m) e,T x = T [m] m • µ (m)
e,x for every x and it follows that the measure defined by the above integral is invariant under T

4.4.

A positivity result. In this section, again (X, µ, T ) is an ergodic inverse limit of k-step nilsystems, with base point e ∈ X.

In the next proposition, the notation ǫ = ǫ 1 . . . ǫ m ∈ {0, 1} m is more convenient that ǫ ⊂ [m]. We recall that 00 . . . 0 ∈ {0, 1} m corresponds to ∅ ⊂ [m] and that 00 . . .

01 ∈ {0, 1} m corresponds to {m} ⊂ [m]. For ǫ ∈ {0, 1} m+1 , ǫ 1 . . . ǫ m corresponds to ǫ ∩ [m]. Proposition 4.3. Let f ǫ , ∅ = ǫ ∈ {0, 1} m , be 2 m -1 bounded measur- able real functions on X. Then ǫ∈{0,1} m+1 ǫ =00...0 ǫ =00...01 f ǫ 1 ...ǫm (x ǫ ) dµ (m+1) e,e (x)
≥ ǫ∈{0,1} m ǫ =00...0 f ǫ (x ǫ ) dµ (m) e (x) 2 .
Proof. We first reduce the general case to that of an ergodic k-step nilsystem. If (X, µ, T ) is an inverse limit of an increasing sequence of k-step ergodic nilsystems, then the spaces X , are the inverse limits of the corresponding objects associated to each of the nilsystems in the sequence of nilsystems converging to X. Thus it suffices to prove the proposition when (X, T, µ) is an ergodic k-step nilsystem. We write X = G/Γ as usual.

The groups

G (m) e , Γ (m)
e , G is the Haar measure of this nilmanifold. It is convenient to identify X [m+1] with X [m] × X [m] , writing a point x ∈ X [m+1] as x = (x ′ , x ′′ ), where

x ′ = (x ǫ 1 ...ǫm0 : ǫ ∈ {0, 1} m ) and x ′′ = (x ǫ 1 ...ǫm1 : ǫ ∈ {0, 1} m ) . The diagonal map ∆ (m) X : X [m] → X [m+1] is defined by ∆ (m) X (x) = (x, x), that is, for x ∈ X [m] and ǫ ∈ {0, 1} m+1 , ∆ (m) X (x) ǫ = x ǫ 1 ,...,ǫm . We remark that ∆ (m) X (X (m) e ) ⊂ X (m+1) e,e
. We use similar notation for elements of G [m+1] and define the diagonal map ∆ (m)

G : G [m] → G [m+1] . We have ∆ (m) (G (m) e ) ⊂ G (m+1)
e,e and, for every g = (g ′ , g ′′ ) ∈ G e,e = (g, hg)

: g ∈ G (m) e , h ∈ G (m) * .
It follows that

X (m+1) e,e = (x, h • x) : x ∈ X (m) e , h ∈ G (m+1) * .
For every x ′ ∈ X (m) , set

V x ′ = {x ′′ ∈ X (m) : (x ′ , x ′′ ) ∈ X (m+1)
e,e } .

For x ∈ X (m) and g ∈ G Indeed, this image is supported on ν g•x and is invariant under

G (m) * , since G (m) * is normal in G (m) e .
We claim that (10)

µ (m+1)
e,e = δ x × ν x dµ (m) e (x) .

The measure on X

(m+1) e,e
defined by this integral is invariant under translation by elements of the form (1 [m] , h) with h ∈ G (m) *

(note that each δ x × ν x is invariant under such translations). By (9), the measure defined by this integral is also invariant under translation by (g, g) for g ∈ G By (9) again,

ν h•x = ν x for h ∈ G (m) * . Let F denote the σ-algebra of G (m) * -invariant functions. For every bounded Borel function F on X (m) e , (11)
F dν x = E(F | F )(x) µ (m) e -a.e.
To see this, we note that the function defined by this integral is invariant under translation by G (m) * and thus is F -measurable. Conversely, if F is F -measurable, then for µ (m) e almost every x, it coincides ν xalmost everywhere with a constant and so the integral is equal almost everywhere to F (x).

Thus for a bounded Borel function F on X (m) e , using (10) and (11), we have that

F (x ′ )F (x ′′ ) dµ (m+1) e,e (x) = F (x ′ ) F (x ′′ ) dν x ′ (x ′′ ) dµ (m) e (x ′ ) = F • E(F | F ) dµ (m) e = E(F | F ) 2 dµ (m) e ≥ F dµ (m) e 2 .

The measures µ

(m,r) e

. In this section again, (X, µ, T ) is an ergodic inverse limit of k-step nilsystems, with base point e ∈ X. Let m and r be integers with 0 ≤ r < m.

Let ∆ m,r : X [m-r] → X [m] be the map given by for x ∈ X [m-r] and ǫ = ǫ 1 , . . . , ǫ m ∈ {0, 1} m , (∆ m,r x) ǫ = x ǫ r+1 ...ǫm .

We define: is the unique probability measure of this set invariant under these transformations. We have ∆ m,r e [m-r] = e [m] , and, for 1

X (m,r) e = ∆ m,r X (m-r)
≤ i ≤ m -r, ∆ m,r • T [m-r] i = T [m] r+i • ∆ m,r . Therefore: X (m,r) e is the closed orbit of the point e [m] ∈ X (m) under the transfor- mations T [m] i for r + 1 ≤ i ≤ m and µ (m,r) e
is the unique probability measure on this set invariant under these transformations.

For example, X r+1) , where ∆ [r] denotes the diagonal of X [r] . µ (r+1,r) e is the product of the Dirac mass at e [r] by the diagonal measure of X [r] .

= {e [r] } × ∆ [r] ⊂ X ( (m,0) e = X (m) e ⊂ X (m) and µ (m,0) e = µ (m) e . X (r+1,r) e
Since the image of µ (m-r) e

under the projections x → x ǫ with ǫ = ∅, are equal to µ, we have that:

The images of µ (m,r) e under the projections

x → x ǫ for ǫ ⊂ [m], ǫ ⊂ [r], are equal to µ. Therefore, if h ǫ , ǫ ⊂ [m], ǫ ⊂ [r], are 2 m -2 r measurable functions on X with |h ǫ | ≤ 1, we have that (14) ǫ⊂[m] ǫ ⊂[r] h ǫ (x ǫ ) dµ (m,r) e (x) ≤ min ǫ⊂[m] ǫ ⊂[r] h ǫ L 1 (µ) .

A convergence result

In this section, we prove the key convergence result (Proposition 5.4). 5.1. Context. We recall our context, as introduced in Sections 3.4 and 3.5.

The system (X, T ) is associated to the subalgebra A of ℓ ∞ (Z), µ is an ergodic invariant probability measure on X, associated to the averages on the sequence I = (I j : j ≥ 1) of intervals.

For every k ≥ 1, let (Z k , µ k , T ) be the factor of order k of (X, µ, T ). We recall that this system is an inverse limit of (k -1)-step nilsystems, both in the topological and the ergodic theoretical senses. The system (Z k , T ) is distal, minimal and uniquely ergodic, and Z k is given with a base point e k . In a futile attempt to keep the notation only mildly disagreeable, when the base point e k is used as a subindex, we omit the subscript k.

We write π k : X → Z k for the factor map. We recall that this map is measurable, and has no reason for being continuous. For ℓ ≤ k, Z ℓ is a factor of Z k , with a factor map π ℓ,k :

Z k → Z ℓ which is continuous and π ℓ,k (e k ) = e ℓ .
We use various different methods of taking limits of averages of sequences indexed by Z r . For example, in Proposition 5.1, we average over any Følner sequence in Z r . In the sequel, we use iterated limits: if a n : n = (n 1 , . . . , n r ) ∈ Z r is a bounded sequence, we define the iterated limsup of a as

Iter limsup | Av I,n 1 ,...,nr a n 1 ,...,nr | = limsup j 1 →∞ . . . limsup jr→∞ 1 |I j 1 | . . . |I jr | n 1 ∈I j 1 ... nr∈I jr a n 1 ,...,nr .
We define the Iter lim Av a n analogously, assuming that all of the limits exist.

An upper bound.

The next proposition is proved in Section 13 of [START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF]: Proposition 5.1. Let (X, µ, T ) be an ergodic system and (Z d , T, ν) be its factor of order d.

Let f ǫ , ǫ ⊂ [d], be 2 d bounded measurable functions on X. For n = (n 1 , . . . , n d ) ∈ Z d , let a n = ǫ⊂[d] f ǫ (T n•ǫ x) dµ(x) and b n = ǫ⊂[d] E(f ǫ | Z d )(T n•ǫ z) dµ d (z) . Then a n -b n converges to zero in density, meaning that the averages of [a n -b n | on any Følner sequence in Z d converge to zero. Lemma 5.2. Let k ≥ 1, 0 ≤ r ≤ d and h ǫ , ǫ ⊂ [d + 1], ǫ ⊂ [r]
, be 2 d+1 -2 r continuous functions on Z k . Then for every δ > 0, there exists C = C(δ) > 0 with the following property:

Let ψ ǫ , ǫ ⊂ [r], be 2 r sequences belonging to B with absolute value ≤ 1. Then the iterated limsup in n 1 , . . . , n r of the absolute value of the averages on I of

A(n) := ǫ⊂[r] ψ ǫ (n • ǫ) ǫ⊂[d+1] ǫ ⊂[r] h ǫ (T n•ǫ x ǫ ) dµ (d+1,r) k e (x)
is bounded by δ + C r∈ǫ⊂[r]
ψ ǫ I,k+r .

Proof. We write n = (m 1 , . . . , m r-1 , p) and m = (m 1 , . . . , m r-1 ). The expression to be averaged can be rewritten as

A ′ (m, p) = ǫ⊂[r-1] ψ ǫ (m•ǫ)• r∈ǫ⊂[r] ψ ǫ (m•ǫ+p)• ǫ⊂[d+1] ǫ ⊂[r] f ǫ (T m•ǫ+pǫr x ǫ ) dµ (d+1,r) k e (x) .
For m ∈ Z r-1 , we write

Φ m (p) = r∈ǫ⊂[r] ψ ǫ (m • ǫ + p) = r∈ǫ⊂[r] σ m•ǫ ψ ǫ (p)
where σ is the shift on ℓ ∞ (Z). For x ∈ Z (d+1,r) k

, we also write

H(x) = ǫ⊂[d+1] ǫ ⊂[r] h ǫ (x ǫ )
and for every δ > 0, we let C = C(δ) be associated to this continuous function on X (d+1,r) k as in Proposition 3.5. We have

r∈ǫ⊂[r] ψ ǫ (m • ǫ + p) • ǫ⊂[d+1] ǫ ⊂[r] h ǫ (T m•ǫ+pǫr x ǫ ) = Φ m (p)H T [d+1]p r (T [d+1]m 1 1 . . . T [d+1]m r-1 r-1 x)
and thus

limsup j Av p∈I j r∈ǫ⊂[r] ψ ǫ (m • ǫ + p) • ǫ⊂[d+1] ǫ ⊂[r] h ǫ (T m•ǫ+pǫr x ǫ ) ≤ δ + C Φ m I,k+1
for every m and every x ∈ Z .

By a result in [START_REF] Host | Uniformity norms on ℓ ∞ and applications[END_REF], the last limsup is actually a limit and is bounded by

r∈ǫ⊂[r]
ψ ǫ I,k+r .

5.3. Iteration. Proposition 5.3. Let k ≥ 1, 0 ≤ r ≤ d and h ǫ , ǫ ⊂ [d + 1], ǫ ⊂ [r],
be 2 d+1 -2 r bounded measurable functions on Z k . Let φ ǫ , ǫ ⊂ [r], be 2 r continuous functions on X. For n ∈ Z r , define

A(n) = ǫ⊂[r] ψ ǫ (n • ǫ) ǫ⊂[d+1] ǫ ⊂[r] h ǫ (T n•ǫ x ǫ ) dµ (d+1,r) k e (x)
and

B(n) = ǫ⊂[r-1] φ ǫ (T n•ǫ x 0 )• r∈ǫ⊂[r] E(φ ǫ | Z k+r-1 )(x ǫ ) • ǫ⊂[d+1] ǫ ⊂[r] h • p k+r-1,k (x ǫ ) dµ (d+1,r-1)
k+r-1 e (x) .

Then the iterated limit of the averages of A(n) -B(n) is zero.

Proof. We remark that B(n) depends only on n 1 , . . . , n r-1 . By ( 14), it suffices to prove the result in the case that the functions f ǫ are continuous. We can also assume that |φ ǫ | ≤ 1 for every ǫ ⊂ [r].

Let δ > 0 be given and let C be as in Lemma 5.2. For each ǫ with r ∈ ǫ ⊂ [d + 1], let φ ǫ be a continuous function on Z k+r-1 with | φ ǫ | ≤ 1, such that E(φ ǫ | Z k+r-1 )φ ǫ is sufficiently small. We have that ( φ ǫ (T n e k+r-1 ) : n ∈ Z) -(φ ǫ (T n x 0 ) : n ∈ Z) I,k+r ≤ δ/2 r-1 C for every ǫ. This follows from Proposition 3.6. By Lemma 5.2 the iterated limsup of the absolute value of the averages on I of

A(n)- ǫ⊂[r-1] φ ǫ (T n•ǫ x 0 )• r∈ǫ⊂[r] φ ǫ (T n•ǫ e k+r-1 )• ǫ⊂[d+1] ǫ ⊂[r] h ǫ (T n•ǫ x ǫ ) dµ (d+1,r) k e (x)
is bounded by 2δ. We rewrite the second term in this difference as

ǫ⊂[r-1] φ ǫ (T n•ǫ x 0 ) • r∈ǫ⊂[r] φ ǫ (T n•ǫ e k+r-1 )• ǫ⊂[d+1] ǫ ⊂[r] h ǫ • p k+r-1,r (T n•ǫ x ǫ ) dµ (d+1,r) k+r-1 e (x)
and remark that the first product in this last expression depends only on n 1 , . . . , n r-1 . By definition of the measures, the averages in n r on I of the above expression converges to

ǫ⊂[r-1] φ ǫ (T n•ǫ x 0 ) • r∈ǫ⊂[r] φ ǫ (x ǫ ) • ǫ⊂[d+1] ǫ ⊂[r] h ǫ • p k+r-1,k (x ǫ ) dµ (d+1,r-1) k+r-1 (x)
.

By (14) again, for every n 1 , . . . , n r-1 the difference between this expression and B(n) is bounded by δ.

The announced result follows.

Proposition 5.4. Let k ≥ 1 and let f ǫ , ǫ ⊂ [d + 1], ǫ = ∅, be 2 d+1 -1 continuous functions on X. Then the iterated averages for n = (n 1 , . . . , n d , n d+1 ) ∈ Z d+1 on I of (15)

∅ =ǫ⊂[d+1] f ǫ (T n•ǫ x 0 ) converge to (16) ∅ =ǫ⊂[d+1] E(f ǫ | Z d )(x ǫ ) dµ (d+1) d e (x) .
Proof. For notational convenience we define f ∅ to be the constant function 1.

By [START_REF] Bergelson | Piecewise-Bohr sets of integers and combinatorial number theory[END_REF], the averages in n d+1 of (15) converge to (17)

∅ =ǫ⊂[d] f ǫ (T n•ǫ x 0 ) • ǫ⊂[d+1] ǫ ⊂[d] f ǫ (T n•ǫ x) dµ(x)
and it remains to show that the iterated averages in (n 1 , . . . , n d ) of this expression converge to (16). By Proposition 5.1, the difference between the quantity (17) and

A(n) := ǫ⊂[d] f ǫ (T n•ǫ x 0 ) • ǫ⊂[d+1] ǫ ⊂[d] E(f ǫ | Z d )(T n•ǫ x) dµ d (x)
converges to zero in density and we are reduced to study the iterated convergence of the averages of A(n).

We apply Proposition 5.3 with k = d and r = d and left with studying the iterated limit of the averages in n 1 , . . . , n d-1 of

ǫ⊂[d-1] f ǫ (T n•ǫ x 0 )• d∈ǫ⊂[d] E(f ǫ | Z 2d-1 )(x ǫ )• ǫ⊂[d+1] ǫ ⊂[d] E(f ǫ | Z d )•p 2d-1,d (x ǫ ) dµ (d+1,d-1) 2d-1 e (x) .
After dr steps, we are left with the iterated limit of the averages in n 1 , . . . , n r of an expression of the form

ǫ⊂[r] f ǫ (T n•ǫ x 0 ) • ǫ⊂[d+1] ǫ ⊂[r] E(f ǫ | Z ℓ(ǫ) ) • p k,ℓ(ǫ) (x ǫ ) dµ (d+1,r) k e (x) ,
where k = k(r) ≥ d is an integer and where for every ǫ, d ≤ ℓ(ǫ) ≤ k. Finally, after d steps, we have that the iterated limit of the expression (17) exists and is equal to

∅ =ǫ⊂[d+1] E(f ǫ | Z ℓ(ǫ) ) • p k,ℓ(ǫ) (x ǫ ) dµ (d+1) k e (x) ,
where k is an integer and d ≤ ℓ(ǫ) ≤ k for every ǫ.

By Proposition 4.1, the measure µ

(d+1) k e
is relatively independent with respect to its projection µ . For every ǫ,

E E(f ǫ | Z ℓ(ǫ) ) • p k,ℓ(ǫ) | Z d = E(f ǫ | Z d )
and we have that the above limit is equal to (16).

Positivity

In this Section, A, X, µ, I = (I j : j ≥ 1), . . . are as in Sections 3.4 and 3.5. Given a sequence of intervals (J k : k ≥ 1) in Z whose lengths tend to infinity, we assume that for each j ≥ 1, there exists some k = k(j) such that the interval I j is included in J k .

We simplify the notation: we write Z instead of Z d , ν instead of µ d , e instead of e d . If f is a function on X, f = E(f | Z).

6.1. Positivity. Lemma 6.1. Let B ⊂ Z be such that 1 B ∈ A and let f be the continuous function on X associated to this set:

f (T n x 0 ) = 1 B (n) .

Let m ≥ 1 be an integer and let h

ǫ , ∅ = ǫ ⊂ [m], be 2 m -1 nonnegative bounded measurable functions on Z. Assume that ∅ =ǫ⊂[m] h ǫ (x ǫ ) dν (m) e (x) > 0 and that Z \ B is not a PW-Nil d Bohr 0 . Then f (x {m+1} ) • ǫ⊂[m+1] ǫ =∅,{m+1} h ǫ∩[m] (x ǫ ) dν (m+1) e (x) > 0 . Proof. By Proposition 4.3, ǫ⊂[m+1] ǫ =∅,{m+1} h ǫ∩[m] (x ǫ ) dν (m+1) e,e (x) > 0 . For z ∈ Z, define H(z) = ǫ⊂[m+1] ǫ =∅, ǫ ={m+1} h ǫ∩[m] (x ǫ ) dν (m+1) e,z (

x) .

We have that δ := H(e) > 0 and, by Proposition 4.2, H is continuous on Z d . Therefore, the subset

Λ = {n ∈ Z : H(T n e) > δ/2} is a Nil d Bohr-set.
By the same proposition,

f(x {m+1} ) • ǫ⊂[m+1] ǫ =∅,{m+1} h ǫ∩[m] (x ǫ ) dν (m+1) e = f (z)H(z) dν(z) . By Proposition 3.3, this last integral is equal to lim Av I f (T n x 0 )H(T n e) ≥ δ 2 limsup j 1 |I j | |Λ ∩ B ∩ I j | .
If this limsup is equal to zero, then there exist arbitrarily long intervals J ℓ such that Λ ∩ B ∩ J ℓ = ∅ and thus the set Z \ B contains Λ ∩ J ℓ for all ℓ. Therefore Z \ B is a PW-Nil g Bohr-set, hence a contradiction.

∅ =ǫ⊂[m] f (x ǫ ) dν (m) e (x) > 0 . (18)
Proof. We remark first that f dµ > 0. Indeed, if this integral is zero, then the density of the set B in the intervals I j converges to 0 and Z \ B contains arbitrarily long intervals, a contradiction. We show (18) by induction. We have that ν

e = δ e × ν and thus f (x 1 ) dν

d e (x) = f dν = f dµ > 0 .

Assume that (18) holds for some m ≥ 1. Then Lemma 6.1 applied to h = f shows that it holds for m + 1.

6.2. And now we gather all the pieces of the puzzle. Recall that if E is a finite subset of N, S(E) is the set consisting in all sums of distinct elements of E (the empty sum is not considered). A subset

A of Z is a S * m -set if A ∩ S(E) = ∅ for every subset E of N with m elements.
We prove Theorem 2.6:

Theorem. Let A be a S * d+1 set. Then A is a PW-Nil d Bohr-set. Proof. Let B = Z \ A, A a subalgebra of ℓ ∞ (Z) containing 1 B
and X, µ, I, . . . are as above. The continuous function f on X is associated to 1 B and we use the same notation as above.

Assume that A is not a PW-Nil d Bohr-set. By Corollary 6.2,

∅ =ǫ⊂[d+1] f(x ǫ ) dν (d+1) e (x) > 0
and by Proposition 5.4, this integral is equal to the iterated limit of the averages in n = (n 1 , . . . , n d+1 ) of

∅ =ǫ⊂[d+1] f (T n•ǫ x 0 ) .
This product is nonzero if and only if S({n 1 , . . . , n d+1 }) ⊂ B. But the complement A of B in Z is a S * d+1 -set (recall that the n i belong to some of the intervals I j ), and so this can not happen.

Proof of Theorem 2.10

We now prove Theorem 2.10 (recall that Theorem 2.8 is a particular case of this theorem):

Theorem. Every SG * d -set is a PW-Nil d Bohr 0 -set. 7.
1. The method. The proof is by contradiction. In this section, d ≥ 1 is an integer and A is a subset of the integers. We assume that A is not a PW-Nil d Bohr 0 -set and by induction, we build an infinite sequence P = (p j : j ≥ 1) such that A ∩ SG d (P ) = ∅.

Let A be a subalgebra of ℓ ∞ (Z) containing 1 A and let X, µ, . . . and the sequence of intervals I = (I j : j ≥ 1) be as in Sections 3.4 and 3.5. We have the same conventions as in the preceding section for the intervals I j .

We write B = Z \ A and let f be the continuous function on X associated to 1 B (see Section 3.4):

f (T n x 0 ) = 1 if n ∈ B ; 0 otherwise.
As in Section 6, we simplify the notation: we write Z instead of Z d , ν instead of µ d , and e instead of e d . If f is a function on X, f = E(f | Z).

In this section, it is more convenient to index points of X [d] by {0, 1} d instead of by P([d]). Thus a point x ∈ X [d] is written x = (x ǫ : ǫ ∈ {0, 1} d ).

For every j ≥ 1, by induction we build 2 d -1 continuous nonnegative functions h We start by setting all of the functions h (0) ǫ , 00 . . . 0 = ǫ ∈ {0, 1} d , to be equal to f . By Corollary 6.2 applied with m = d and rewritten in the current notation, we have that property (19) is satisfied for i = 0. 7.2. Iteration. Assume j ≥ 1 and that property (19) is satisfied for j -1.

By Proposition 4.3, ǫ∈{0,1} d+1 ǫ =00...0,ǫ =00...01 h (j-1) ǫ 1 ...ǫ d (x ǫ ) dν (d+1)

e,e (x) > 0 .

By Lemma 6.1, rewritten in our current notation, we have that f (x 00...01 ) • ǫ∈{0,1} d+1 ǫ =00...0,ǫ =00...01 h (j-1) ǫ 1 ...ǫ d (x ǫ ) dν (d+1) e (x) > 0 .

For convenience, we write h The averages in p over the intervals I of this expression converge to the limit (20), which is positive. Thus there exists some p (belonging to some I i ) such that this expression is positive. Choosing p j to be this p, for 00 . . . 0 = ǫ ∈ {0, 1} d , we define h (j) ǫ = g p j ,ǫ = h (recall that h We have the inductive relations φ 0,k = f for 1 ≤ k ≤ d ; φ j-1,1 (T p j x 0 ) > 0 ; for 1 ≤ k < d, φ j,k = φ j-1,k+1 • T p j φ j-1,1 ; φ j,d = f • T p j φ j-1,1 .

By induction, φ j,1 ≤ φ j,2 ≤ • • • ≤ φ j,d ≤ f . Moreover, we deduce the following relations between the functions φ j,1 :

for 1 ≤ j < d, φ j,1 = f • j k=1 T p j-k+1 φ j-k,1

for j ≥ d, φ j,1 = f • d k=1 T p j-k+1 φ j-k,1 .

For every j, there is a finite set E j of integers with φ j,1 = q∈E j T q f . We have that E 0 = {0} and the E j satisfy that relations for 1 ≤ j < d, E j = {0} ∪ (E j-1 + p j ) ∪ (E j-2 + p j-1 ) ∪ . . . ∪ (E 0 + p 1 ) , for j ≥ d, E j = {0} ∪ (E j-1 + p j ) ∪ (E j-2 + p j-1 ) ∪ . . . ∪ (E j-d + p j-d+1 ) .

By induction, E j consists in all sums of the form ǫ 1 p 1 + • • • + ǫ j p j where ǫ i ∈ {0, 1} for all i, and, after the first occurrence of 1, there can be no block of d consecutive 0's. By induction, each function φ j,1 only takes on the values of 0 and 1 and corresponds to a subset B j of the integers and we have

B j = q∈E j (B -q) .
For every j, since φ j-1,1 (T p j x 0 ) > 0, we have that p j ∈ B j-1 and thus that E j-1 + p j ⊂ B.

We conclude that all sums of the form ǫ 1 p 1 +• • •+ǫ k p k with ǫ i ∈ {0, 1} for all i belong to B, provided the ǫ i are not all equal to 0 and that the blocks of consecutive 0's between two 1's have length < d. In other words, B ⊃ SG d ({p j : j ≥ 1}) and we have a contradiction.

We note that at each step in the iteration, we have infinitely many choices for the next p. In particular, we can take the p j tending to infinity as fast as we want. More interesting, in the construction we can choose a different permutation of coordinates at each step. This gives rise to different, but related, structures, which do not seem to have any simple description.

1

  If A ⊂ Z, the upper Banach density d * (A) is defined to be limsup bn-an→∞ |A ∩ [a n , b n ]| b na n .

Theorem 2 .

 2 10. Every SG * d -set is a PW-Nil d Bohr 0 -set. If |P | = d + 1, then SG d (P ) = S(P ) and thus Theorem 2.10 generalizes Theorem 2.6.

Question 2 .

 2 11. Is every Nil d Bohr 0 -set an SG * d -set? As our characterizations of the sets SG d and the class SG * d are complicated, we ask the following: Question 2.12. Find an alternate description of the sets SG d and of the class SG * d .

3. 4 . 3 .

 43 Notation. In the sequel, when a = (a n : n ∈ Z) is a bounded sequence, we write lim Av n,I a n = lim j→+∞ 1 |I j | n∈I j a n if this limit exists, and set limsup Av n,I a n = limsup j→+∞ 1 |I j | n∈I j a n .

  ν, T ) be an inverse limit of k-step nilsystems and let p : X → Y be a factor map. Then ν(m) e is the image of µ (m) e under p [m] : X [m] → Y [m] . d) Let (Y,ν, T ) be the (m -1)-step factor of X and p : X → Y be the factor map. Then the measure µ (m) e is relatively independent with respect to ν (m) e , meaning that when f

  is easy to check that G (m) e is spanned by the connected component of its identity and the elements τ [m] i , 1 ≤ i ≤ m. Moreover, by using the above description of G (m) e , it is not difficult to check that the action induced by T [m] i , 1 ≤ i ≤ m on the compact abelian group G e is ergodic. By a classical criteria [8], the action of the transformations T [m] i on X (m) e is ergodic and thus minimal. In particular, X (m) eis the closed orbit of the point e[m] under these transformations. This proves a).We now prove d). The (m -1)-step nilfactor (Y, ν, T ) of (X = G/Γ, µ, T ) is Y = G/ΓG m endowed with its Haar measure.For every ǫ ⊂ [m] with ǫ = ∅ and every w ∈ G m , we have w {ǫ} ∈ G (m) e and thus the Haar measure µ

Proposition 4 . 2 .

 42 For each x ∈ X, there exists a measure µ (m) e,x , concentrated on X (m) e,x , such that i) The image of µ (m) e,x under each projection

  • e [m] = X (m)e,e . It follows that Γ(m) e,e := Γ[m] ∩ G (m) for the Haar measure of this nilmanifold. Let F = {ǫ ⊂ [m] : m ∈ ǫ}. We recall that for g ∈ G, g F ∈ G (m) is defined by(g F ) ǫ = g if m ∈ ǫ 1 otherwise.By definition of the sets X is normal in G (m) , the image of the measure µ (m) e,e under g F is invariant under G (m) e,e . Moreover, if g, h ∈ G satisfy g • e = h • e, then we have that g = hγ for some γ ∈ Γ. Since γ F • e [m] = e [m] and by normality of G (m) e,e again, the measure µ (m) e,e is invariant under γ F and thus the images of µ (m)

  we have that g ′ and g ′′ belong toG (m) e ; in other words, G define G (m) * = {g ∈ G (m) e : (1 [m] , g) ∈ G (m+1)

  the image of ν x under translation by g is equal to ν g•x .

  this measure is invariant under G (m+1) e,e . Since it is supported on X (m+1) e,e , it is equal to the Haar measure µ (m+1) e,e of this nilmanifold. The claim is proven.

  of µ (m-r) e under ∆ m,r . (13) Recall that X (m-r) e is the closed orbit of e [m-r] under the transformations T [m-r] i for 1 ≤ i ≤ m-r and that µ (m-r) e

j≤

  Av p∈I j A ′ (m, p) ≤ δ + C Φ m I,k+1 for every m. Therefore Iter limsup Av I,n 1 ,...,nr A(n) Iter limsup Av I,n 1 ,...,n r-1 lim j Av p∈I j A ′ (m, p) ≤ δ + C Iter limsup Av I,m 1 ,...,m r-1 Φ m I,k+1 ≤ δ + C Iter limsup Av I,m 1 ,...,m r-1 Φ m

ǫ

  , 00 . . . 0 = ǫ ∈ {0, 1} d , on X satisfying (19)

  ǫ∈{0,1} d ǫ =00...0 h (j) ǫ (x ǫ ) dν (d) e (x) > 0 .

(j- 1 )

 1 00...0 = f and rewrite this equation as (20) ǫ∈{0,1} d+1 ǫ =00...0 h (j-1) ǫ 1 ...ǫ d (x ǫ ) dν (d+1) e (x) > 0 .By Proposition 5.4, this last integral is the iterated limit of the averages for n = (n 1 , . . . , n d+1 ) of ǫ∈{0,1} d+1 ǫ =00...0h (j-1) ǫ 1 ...ǫ d (T n•ǫ x 0 ) .We make a change of indices, writing elements of Z d+1 as (p, n 1 , . . . , n d ) and setting n = (n 1 , . . . , n d ). Elements of {0, 1} d+1 are written as ηǫ 1 . . . ǫ d with η ∈ {0, 1} and we set ǫ = ǫ 1 . . . ǫ d . The last product becomes:h (j-1) 100...0 (T p x 0 ) ǫ∈{0,1} d ǫ =00...0 h (j-1) 0ǫ 1 ...ǫ d-1 • T p h (j-1) 1ǫ 1 ...ǫ d-1 (T n•ǫ x 0 ) .For ǫ ∈ {0, 1} d , ǫ = 00 . . . 0, and for p ∈ Z, setg p,ǫ = h (j-1) 0ǫ 1 ...ǫ d-1 • T p h (j-1) 1ǫ 1 ...ǫ d-1and rewrite the last expression ash (j-1) 100...0 (T p x 0 ) ǫ∈{0,1} d ǫ =00...0 g p,ǫ (T n•ǫ x 0 ) .By Proposition 5.4 again, the iterated limit of the averages in n 1 , . . . , n d converges toh (j-1) 100...0 (T p x 0 ) ǫ∈{0,1} d ǫ =00...0 E g p,ǫ | Z d-1 (x ǫ ) dµ .0 (T p x 0 ) ǫ∈{0,1} d ǫ =00...0 g p,ǫ (x ǫ ) dµ (d) e (x)because the measure µ (d) e is relatively independent with respect to µ (m) d-1 e (see Proposition 4.1, part (d)).

1

 1 ...ǫ d-1 • T p j h (j-1) 1ǫ 1 ...ǫ d-1

(j- 1 )

 1 00...0 = f ). Since (19) is valid with h .0 (T p j x 0 ) > 0 . 7.2.1. Interpreting the iteration. By induction, it follows that for every j ≥ 0, the functions h j,ǫ , 00 . . . 0 = ǫ ∈ {0, 1} d , only depend on the first nonzero digit of ǫ:h j,ǫ = φ j,k if ǫ 1 = • • • = ǫ k=1 = 0 and ǫ k = 1 .

The theory of representations does not help us, as the interesting representations of a nilpotent Lie group are infinite dimensional.

We are forced to use different notation from that in[START_REF] Host | Nonconventional ergodic averages and nilmanifolds[END_REF], as otherwise the proliferation of indices would be uncontrollable.

Acknowledgment. Hillel Furstenberg introduced us to this problem and we thank him for his encouragement, as well as for helpful comments on a preliminary version of this article.


Hosoon Choi

Piyali Chatterjee

Munok Hwang

Eric Lichtfouse

Virender K Sharma
email: vsharma@tamu.edu

Chetan Jinadatha
email: chetan.jinadatha@va.gov

The viral phoenix: enhanced infectivity and immunity evasion of SARS-CoV-2 variants

SARS-CoV-2 pandemic continues with emergence of new variants of concerns. These variants are fueling the third and fourth waves of pandemic across many nations. Here we describe the new emerging variants of SARS-CoV-2 and why they have enhanced infectivity and possess the ability to evade immunity.

The coronavirus disease that started in 2019, named COVID-19, has induced more than 210 million cases and 4 million deaths in September 2021, following the infection by the severe acute respiratory syndrome coronavirus-2 (SARS-CoV-2, [START_REF] Sharma | Environmental chemistry is most relevant to study coronavirus pandemics[END_REF][START_REF] Sharma | COVID-19 epidemiologic surveillance using wastewater[END_REF]. Actually, many scientific issues remain unanswered such as the virus origin, transmission and adaptation, and the role of environmental factors such as pollution (Chen et al. 2021a, b;Choi et al. 2021a, b;[START_REF] Dai | Smarter cures to combat COVID-19 and future pathogens: a review[END_REF][START_REF] Khan | COVID-19 transmission, vulnerability, persistence and nanotherapy: a review[END_REF]Roviello and Roviello 2020;[START_REF] Paital | Air pollution by NO 2 and PM2.5 explains COVID-19 infection severity by overexpression of angiotensinconverting enzyme 2 in respiratory cells: a review[END_REF][START_REF] Sun | Open defecation and squat toilets, an overlooked risk of fecal transmission of COVID-19 and other pathogens in developing communities[END_REF][START_REF] Ufnalska | Unanswered issues related to the COVID-19 pandemic[END_REF]. Despite the design and massive administration of several vaccines, the pandemic continues with emergence of new variants of concerns, similarly to the Greek mythological phoenix, an immortal bird that cyclically regenerates by obtaining new life by arising from the ashes of its predecessor. These variants are fueling the third and fourth waves of the pandemic across many nations. Here, we describe the new emerging variants of SARS-CoV-2 and why they have enhanced infectivity and possess the ability to evade immunity.

Variants of concern

SARS-CoV-2 has slower mutation rate than other RNA viruses due to the presence of the proofreading 3'-5' exonuclease nsp14 [START_REF] Robson | Coronavirus RNA proofreading: molecular basis and therapeutic targeting[END_REF]). However, many viral genome replications that occur during the pandemic are increasing the genetic diversity of SARS-CoV-2. Most mutations are detrimental to virus fitness, yet mutations beneficial to virus survival and transmission are selected and accumulated over time to give rise to variants with enhanced transmissibility, hence leading to a rapid rise of worldwide infections. Variant of concerns are variants for which there is significant evidence to have increased transmissibility, disease severity, and immune evasion potency. Currently, four variants of SARS-CoV-2, alpha-lineage B.1.1.7, beta-lineage B.1.351, gamma-lineage P.1, and delta-lineage B.1.617.2 are recognized as variants of concern by the World Health Organization (WHO) and the Centers for Disease Control and Prevention (CDC) (Table 1). The two main features that control the natural selection of SARS-CoV-2 are the extreme variability of disease severity and the temporal patterns of viral shedding. Patients with SARS-CoV-2 infection undergo a range of clinical symptoms, from no symptoms to critical illness and death [START_REF] Garcia | Immune response, inflammation, and the clinical spectrum of COVID-19[END_REF]. About one fourth of infections are asymptomatic cases [START_REF] Alene | Magnitude of asymptomatic COVID-19 cases throughout the course of infection: a systematic review and metaanalysis[END_REF]. Theoretically, direct selection pressure against virulence of SARS-CoV-2 is likely very weak because of the wide symptomatic spectrum and the fact that the peak viral load and transmissibility occur days before symptom onset. (*) = detected in some sequences but not all a Phylogenetic Assignment of Named Global Outbreak (PANGO) Lineages is a software tool developed by members of the Rambaut Lab. The associated web application was developed by the Centre for Genomic Pathogen Surveillance in South Cambridgeshire and is intended to implement the dynamic nomenclature of SARS-CoV-2 lineages, known as the PANGO nomenclature b Nextstrain, a collaboration between researchers in Seattle, USA and Basel, Switzerland, provides an open-source tool for visualizing the genetics of outbreaks. The goal is to support public health surveillance by facilitating understanding of the spread and evolution of pathogens Nextstrain Clade (b) WHO Classification Pango Lineage (a) Sequence

Evolving viral keys unlock cell entrance

Amid rapid spread of COVID-19, transmissibility is a major selective force for SARS-CoV-2. Indeed, higher viral loads are increasing the transmission rate and, in turn, more transmission can be accompanied by more severe illness. The transmission rate is mainly controlled by the recognition and binding of the virus to host cells.

The spike (S) protein of SARS-CoV-2 plays a key role in the receptor recognition and cell membrane fusion process [START_REF] Hoffmann | SARS-CoV-2 cell entry depends on ACE2 and TMPRSS2 and Is blocked by a clinically proven protease inhibitor[END_REF][START_REF] Luchini | Lipid bilayer degradation induced by SARS-CoV-2 spike protein as revealed by neutron reflectometry[END_REF]). In the host cell membrane, the angiotensin-converting enzyme 2 (ACE2) has been identified as the viral receptor (Fig. 1). The S glycoprotein mediates the entry of the virus into host cells through its receptor-binding domain (RBD) and ACE2 interaction [START_REF] Wan | Receptor recognition by the novel coronavirus from Wuhan: an analysis based on decade-long structural studies of SARS coronavirus[END_REF][START_REF] Yan | Structural basis for the recognition of SARS-CoV-2 by full-length human ACE2[END_REF]. The crystal structure of the S glycoprotein bound to ACE2 has been determined [START_REF] Lan | Structure of the SARS-CoV-2 spike receptor-binding domain bound to the ACE2 receptor[END_REF]. SARS-CoV-2 has an almost identical binding complex as SARS-CoV-1, the coronavirus strain that caused the death of 9% of patients in 2003, but SARS-CoV-2 displays a tenfold higher binding affinity [START_REF] Li | Structure of SARS coronavirus spike receptor-binding domain complexed with receptor[END_REF]. Indeed, in the SARS-CoV-2 receptor-binding domain, hydrogen bonds are formed between amino acid residues of host ACE2 and the following sites of the virus spike protein: K417, G446, Y449, N487, Y489, Q493, T500, N501, G502, and Y505. A salt-bridge connection has also been identified between the virus K417 site and the D30 site of host ACE2 [START_REF] Lan | Structure of the SARS-CoV-2 spike receptor-binding domain bound to the ACE2 receptor[END_REF]). Here, mutations in the receptor-binding domain of the spike protein can result in higher affinity to ACE2 and can thus lead to an increase in infectivity and transmissibility of the virus. For instance, N501 is mutated in alpha, beta, and gamma variants. Mutations in K417 are found in beta, gamma, and delta plus variant-lineage AY.1.

Zoonosis

Human-to-animal and animal-to-human transmission, favored by pollution, are very likely routes for the pandemic origin and propagation, and for vaccine failure (He et al. 2021a,b). Research on interspecies transmission reveals that amino acid residues play major roles in the virus transmissibility. For instance, the Danish Cluster 5 variant has spread from human to minks as a reverse-zoonotic transmission in Denmark then followed by minks to human transmission [START_REF] Oreshkova | SARS-CoV-2 infection in farmed minks, the Netherlands[END_REF][START_REF] Munnink | Transmission of SARS-CoV-2 on mink farms between humans and mink and back to humans[END_REF]. This variant contains the Y453F mutation in its receptorbinding domain of the S glycoprotein which is associated with a fourfold higher affinity to ACE2 [START_REF] Welkers | Possible hostadaptation of SARS-CoV-2 due to improved ACE2 receptor binding in mink[END_REF].

A mouse adapted SARS-CoV-2 strain carry Q493K, Q498Y and P499T mutations [START_REF] Leist | A mouseadapted SARS-CoV-2 induces acute lung injury and mortality in standard laboratory mice[END_REF]. Tiger and lion were infected with SARS-CoV-2, which contain F456Y or Y505H mutations in the receptor-binding domain, respectively [START_REF] Mcaloose | From people to panthera natural SARS-CoV-2 infection in tigers and lions at the Bronx Zoo[END_REF].

Immune evasion

As worldwide vaccination progresses, the immune evasion potency becomes a stronger selective force in viral evolution. Indeed, the antigenic ability of the virus can change as mutations alter the protein structure. Some mutations can lead to reduced effectiveness of the existing vaccines which were developed against the original strain of the virus. For example, the receptor-binding motif (RBM) is a small patch of about 70 amino acids located in the receptor-binding domain of the host ACE2. In this domain, amino acid residues not only form bonds with the virus spike protein, but also serve as antibody binding epitopes for many neutralizing antibodies. Here, the weakening of some neutralizing antibody binding is attributed to mutations of residues K417 and E484 (Chen et al. 2021a, b;[START_REF] Wise | Covid-19: The E484K mutation and the risks it poses[END_REF]). K417N mutations have been found in beta and delta plus variants. Gamma variant contains a K417T mutation. E484K mutation is found in beta and gamma variants. The N-terminal domain (NTD) is another major target for neutralizing antibodies. Two deletion mutations, 69-70 ∆HV and 144 ∆Y, in the N-terminal domain of the spike protein confer the resistance against the NTD-directed antibodies to alpha variants. The resistance against NTD-targeting antibodies of beta variant largely comes from the R246I mutation. The P26S, L18F, T20N, D138Y, and R190S mutations play similar roles in immune evasion of the gamma variant.

There are heightened concerns about the existing COVID-19 vaccine effectiveness against current and emerging variants of SARS-CoV-2. Potency of vaccine-induced neutralizing antibodies against SARS-CoV-2 variants showed several fold reductions against gamma and delta variants, and about 20-40 fold decreases against beta variant [START_REF] Noori | Potency of BNT162b2 and mRNA-1273 vaccine-induced neutralizing antibodies against severe acute respiratory syndrome-CoV-2 variants of concern: A systematic review of in vitro studies[END_REF]). However, the impact of this neutralization titer reduction on vaccine effectiveness is hard to predict. Although neutralizing antibody levels are predictive of immune protection, the threshold of neutralization for vaccine protection is not known yet [START_REF] Khoury | Neutralizing antibody levels are highly predictive of immune protection from symptomatic SARS-CoV-2 infection[END_REF]. Furthermore, the impact of the other potential protective factors besides neutralizing antibody levels, such as T-cell mediated immunity and memory B-cell, has not been fully understood yet.

Delta variant

SARS-CoV-2 is expected to continuously change. Some variants will emerge and replace existing variants. The Delta variant is currently the subject of the main concern. Delta variant contains L452R and T478K in the receptor-binding motif and P681R in the S1/S2 cleavage site in its spike protein. Delta variants are rapidly spread globally and reached actually more than 90% of new cases in US (https:// covid. cdc. gov/ covid-data-track er/# varia nt-propo rtions). The delta variant is highly contagious. The transmission rate is 2 to 3 times higher than original virus, and the viral load in respiratory tract is about 1,000 times higher than original virus [START_REF] Li | Viral infection and transmission in a large well-traced outbreak caused by the Delta SARS-CoV-2 variant[END_REF]. Delta variants show reduced sensitivity to sera from vaccinated individuals [START_REF] Planas | Reduced sensitivity of SARS-CoV-2 variant delta to antibody neutralization[END_REF]. However, current vaccines are still highly effective against the delta variant [START_REF] Nasreen | Effectiveness of COVID-19 vaccines against variants of concern in Ontario[END_REF][START_REF] Sheikh | SARS-CoV-2 Delta VOC in Scotland: demographics, risk of hospital admission, and vaccine effectiveness[END_REF]. For instance, only modest difference of the vaccine effectiveness between delta and alpha variants was reported when two doses of the BNT162b2 Pfizer-BioNTech or the Oxford-AstraZeneca ChAdOx1 nCoV-19 vaccines were received [START_REF] Bernal | Effectiveness of Covid-19 vaccines against the B.1.617.2 (Delta) variant[END_REF]. However, some vaccinated can not only be infected with Delta variants but also may spread virus to others [START_REF] Brown | Outbreak of SARS-CoV-2 infections, including COVID-19 vaccine breakthrough infections, associated with large public gatherings -Barnstable County, Massachusetts, July 2021[END_REF].

As SARS-CoV-2 spread rapidly, there is a higher chance that new variants can escape immune surveillance, thus rendering current vaccines ineffective. Thorough investigations of variant infection cases and close monitoring of the emergence of new variant are urgently needed. Expediting vaccine deployment and implementing other mitigation measures including mask wearing and social distancing are critical in preventing the emergence and transmission of vaccine escape variants of SARS-CoV-2. Moreover, in the long run, research on alternative cures and prevention measures is definitively needed to outsmart the virus adaptation [START_REF] Dai | Smarter cures to combat COVID-19 and future pathogens: a review[END_REF]).

Table 1

 1 Variants of concern according to the Centers for Disease Control and Prevention, on December 8, 2021


Wafa Ben Hassen

Fabrice Auzanneau

Luca Incarbone

François Peres

Ayeley P Tchangani

On-Line Diagnosis using Orthogonal Multi-Tone Time Domain Reflectometry in a Lossy Cable

Keywords: Solution: wiring networks diagnosis system., Interest: detect, locate and characterize faults., Advantages: maintenance cost reduction (time, resources, etc.), system reliability increase, etc, Cable length increase., Connections number increase., Faults number caused by cables increase, Embedded electronic explosion., Embedded systems complexity increase., «X-by-wire» technology appearance unlimited spectrum (sinc) Communication Signal Test Injection Signal Reception Simulation Results: Lossy Cable Tones Allocation for Sensors S1 Orthogonal Multi-Tone Time Domain Reflectometry

Ensures communication among senors to avoid faults localisation ambiguities

Improves soft faults detection.

OMTDR will be tested on more complex wiring networks.

Reflectometry:

• A wideband test signal is injected down the wire. During its propagation, a part of its energy reflects back to the injection port when it crosses impedance discontinuities.

• The analysis of the received signals, called "reflectogram", gives information about the wire faults (presence, position, type).

S R

  FIGHTER AIRCRAFTTRAIN FPSO

qq

  MCR: Multi-Carrier Reflectometry etry MC CR: Multi Carrier Reflectome Reflectome MCTDR: Multi-Carrier TDR metry FD DR: Frequency Domain Reflectom FD DR: Frequency Domain Reflectom TDR: Time Domain Reflectometry Ø TDR injects a pulse (Gaussian, rectangular, etc.) down the cable under test. Ø FDR uses a set of sinusoidal waves. Ø While STDR uses a Pseudo Noise (PN) code as test signal, SSTDR employs a sine wave modulated PN code. Ø MCR and MCTDR inject a sum of sine waves using Inverse Digital Fourier Transform (IDFT). The 5 67 tone is expressed as: t § Time domain: § Frequency domain: The OFDM signal autocorrelation is: q In discrete domain, the sample of OFDM signal is: Problem of side lobes: unsuitable for reflectometry diagnosis Solution: use butterworth filter with frequency-cut equal to 1

© CEA. All rights reserved


MPH Kalonde Malama
email: kalondemalama@gmail.com

Luis Sagaon-Teyssier

MSPH Rachel Parker

MPH Amanda Tichacek

MPH, DrPH Tyronza Sharkey

MD, MSc William Kilembe

MD, MPH Mubiana Inambao

PhD Matt A Price

MD, PhD Bruno Spire

MD, MPH Susan E Allen

PhD Luis Sagaon Teyssier

Client-Initiated Violence against Zambian Female Sex Workers: Prevalence and Associations with Behaviour, Environment, and Sexual History

Keywords: Physical violence, Female sex workers, Clients, HIV, Zambia

Introduction

With an estimated HIV prevalence of 36.9% (36.2-37.5), female sex workers (FSW) in sub-Saharan Africa are a high-risk group. This vulnerability is due to a combination of sociobehavioural factors such as multiple sexual partners, stigmatisation, and criminalisation [START_REF] Baral | Burden of HIV among female sex workers in low-income and middle-income countries: a systematic review and meta-analysis[END_REF][START_REF] Ngugi | Female sex workers in Africa: Epidemiology overview, data gaps, ways forward[END_REF][START_REF] Scorgie | Socio-Demographic Characteristics and Behavioral Risk Factors of Female Sex Workers in Sub-Saharan Africa: A Systematic Review[END_REF]. The discrimination and lack of legal protection around sex work also render FSW vulnerable to violence from their clients, intimate partners, and the police [START_REF] Deering | A Systematic Review of the Correlates of Violence Against Sex Workers[END_REF].

Violence intersects with HIV both directly and indirectly. When threatened with violence, FSW have less agency to negotiate condom use [START_REF] Jewkes | Intimate partner violence, relationship power inequity, and incidence of HIV infection in young women in South Africa: a cohort study[END_REF]. And during a forced sexual act, perpetrators are unlikely to use a condom [START_REF] Jewkes | Intimate partner violence, relationship power inequity, and incidence of HIV infection in young women in South Africa: a cohort study[END_REF][START_REF] Kouyoumdjian | Intimate partner violence is associated with incident HIV infection in women in Uganda[END_REF]. This results in potential exposure to HIV for both FSW and perpetrators of violence. Structural violence from health institutions can deny FSW access to vital HIV prevention services based on the stigma surrounding sex work [START_REF] Scorgie | We are despised in the hospitals': sex workers' experiences of accessing health care in four African countries[END_REF]World Health Organisation, 2005). FSW who have experienced violence therefore face mental and reproductive health consequences. These include depression, unwanted pregnancy, exposure to sexually transmitted infections (STI), and reluctance to test for STI [START_REF] Sherwood | Sexual violence against female sex workers in The Gambia: a crosssectional examination of the associations between victimization and reproductive, sexual and mental health[END_REF][START_REF] Tounkara | Violence, condom breakage and HIV infection among female sex workers in Benin, West Africa[END_REF][START_REF] Wirtz | Sexual violence, condom negotiation, and condom use in the context of sex work: results from two West African countries[END_REF].

In Zambia, the existing gender norms and unfavourable legal climate render FSW susceptible to violence from their male clients. 43% of Zambian women aged 15-49 report a history of physical violence, the majority of which is committed by men (Central Statistical Office (Zambia), Ministry of Health (Zambia), and ICF International, 2014). FSW in Zambia do not typically have intermediaries and often negotiate directly with clients (Zulu, Ngwenya, Silukena, Phiri, & Chiseni, 2015). Moreover, clients of FSW in Zambia are likely to be men who justify physical violence against women [START_REF] Leclerc | Clients of Commercial Sex Workers in Zambia: Prevalence, Frequency, and Risk Factors[END_REF]. The Zambian penal code criminalises activities related to sex work such as loitering, soliciting and procuring. This could force FSW to transact with clients in concealed and unsafe environments such as vehicles or the client's own home, thus elevating the risk of violence.

In two of the few published studies on sex workers in Zambia, FSW reported encountering physical, sexual, and psychological violence from their male clients [START_REF] Abbott | Female Sex Workers, Male Circumcision and HIV: A Qualitative Study of Their Understanding, Experience, and HIV Risk in Zambia[END_REF][START_REF] Agha | Life-circumstances, working conditions and HIV risk among street and nightclub-based sex workers in Lusaka, Zambia[END_REF]. However, these studies were qualitative and did not measure violence as the primary outcome or model its determinants, which is important for informing safety interventions for FSW. To address this knowledge gap, we conducted a cross-sectional study to determine the prevalence of client-initiated physical violence against Zambian FSW and its associated factors. Our study focuses on commercial sex workers, whose transactions involve a client and an exchange of money for sex.

Methods

Study participants

This cross-sectional analysis draws from baseline data collected from 419 HIV-negative FSW selected from a larger HIV-incidence cohort for high-risk women at the Zambia Emory HIV Research Project (ZEHRP). Enrolment took place between September 2012 and May 2015.

Eligibility criteria included being HIV-negative, aged 18-45, and sexually active. The mean age of FSW in this study was 25, most of whom resided in Ndola (65%) and were educated up to primary school level (48%).

Study procedures

The study was approved by the Emory University Institutional Review Board (Atlanta, USA) and the University of Zambia Research Ethics Committee (Lusaka, Zambia). Seventeen community health workers and 11 peer sex workers recruited FSW from known hotspots of commercial sex work activity in the capital city Lusaka, and the Copperbelt Province city of Ndola. All study participants provided informed consent in either English or one of the local languages-Nyanja or Bemba. Medical staff trained in psychosocial HIV counselling administered a face-to-face quantitative survey collecting information on sociodemographics and HIV risk factors.

Services offered

In addition to risk reduction counselling, participants were offered family planning counselling, condoms, and the choice of a long acting contraceptive method, i.e. intrauterine device or hormonal implant. At baseline, all FSW underwent screening for HIV, syphilis and trichomonas vaginalis. STI treatment was provided at no cost to the participants. All enrolled FSW received transport reimbursement to cover travel to and from the research facility.

During screening and enrolment procedures, participants received food and beverages.

Outcome

FSW were asked if they had ever in their lifetime experienced physical violence from their male clients (yes or no). Client-initiated physical violence was defined as ever being struck or beaten by a paying client. We also asked FSW multiple questions (yes/no) on how they responded after experiencing violence. When we asked women to describe their reaction to experiencing violence, the following responses were recorded: 'ran away', 'called friends or strangers for help', 'stayed calm and negotiated with client', 'called the police', and 'fought back'. These response options were based on pilot study data among FSW at ZEHRP.

Explanatory variables

Potential correlates of violence were chosen from demographic variables including age, education and reason for entering sex work. Sexual history variables accounted for age at sexual debut, coercion at sexual debut, age at entry into sex work and years of experience in sex work. Behavioural indicators were self-reported alcohol use (yes or no) and number of clients in the previous month. Sexual health screening variables were condom use with clients in the previous month, and STI test results. Environmental variables included city of residence and a series of yes or no questions on venues where FSW recruit and have sex with clients.

When surveying FSW on venues where they recruited and had sex with clients, the following responses were permitted: 'bars/nightclubs', 'streets', 'lodges/hotels', 'Where FSW lives', 'Where client lives', 'cars', and 'outside'. These response options were based on pilot study data among FSW at ZEHRP.

Statistical analysis

In the descriptive phase, chi-square tests were performed to illustrate the differences in sample characteristics between FSW who experienced lifetime client-initiated physical violence and their counterparts. Bivariate logistic regression was used to test the association between the outcome variable (client-initiated physical violence) and potential covariates. All variables significant at the p<0.25 level in bivariate analysis were tested in multivariable analysis to ensure purposeful variable selection [START_REF] Bursac | Purposeful selection of variables in logistic regression[END_REF].

Forward stepwise multiple logistic regression was used to construct the final model with adjusted odds ratios (AOR) controlling for age and level of education. Statistical analyses were performed using Stata version 14.2 (StataCorp, College Station, Texas, USA).

Results

The median age at sexual debut was 15 with an interquartile range (IQR) of 14 to 17. The median age of entry into sex work was 18 (IQR: 16-20). FSW mainly recruited their clients from bars (95%) and the street (51%), while lodges (75%) and the client's home (42%) were the most frequented areas where sex took place. The median number of clients seen per month was 10 (IQR: 7-21). Most FSW (57%) reported inconsistent condom use with their clients (Table 1).

The prevalence of ever experiencing client-initiated physical violence was 39% (164/419).

Table 1 displays differences in selected characteristics between FSW who reported experiencing physical violence from their clients versus those who did not. In bivariate analysis, report of violence was significantly higher (p<0.05) amongst FSW who: were physically forced into their first sexual intercourse; recruited clients on the street; provided services at the client's home or in a car; ever used alcohol; and did not always use condoms with clients. Additional factors with borderline significance (P>=0.05 and p<0.1) included living in Lusaka compared with Ndola and recruiting clients in a lodge/hotel compared to not recruiting clients in a lodge/hotel.

When confronted with violence, the majority of FSW ran away (71%) and around a third either calmly negotiated with the client (30%) or called a friend or stranger for help (32%).

Only a minority of FSW called the police (13%) in the aftermath of facing physical violence from their clients, while a smaller proportion (5%) of FSW reported fighting back (Table 2).

Of the 145 FSW who responded: 93 reported only one of these reactions to violence, 32 reported two reactions, 14 reported three reactions and 5 reported four reactions.

In multivariable analysis (Table 3), the odds of client-initiated violence were higher among FSW who were physically forced during their sexual debut (AOR: 2.69, 95% CI: 1.43-5.10) than those whose sexual debut occurred willingly. Having sex where the client lived increased the odds of violence (AOR: 2.94, 95% CI: 1.84-4.74) over not having sex where the client lived. FSW who practised sex work in Lusaka observed higher odds of encountering violence from clients (AOR: 2.13, 95% CI: 1.25-3.62) than their Ndola counterparts. The odds of violence increased significantly for FSW who saw 9-19 (AOR: 1.87, 95% CI: 1.07-3.25) and 20 clients or more (AOR: 2.38, 95% CI: 1.27-4.46) the month prior, compared with FSW who received up to eight clients the previous month. FSW who recruited clients from the street experienced higher odds of violence than those who did not recruit clients from the street (AOR: 1.65, 95% CI: 1.03-2.65).

Discussion

Our study found a 39% prevalence of client-initiated physical violence against FSW. Violence was associated with a history of forced sex, client volume, venue of recruitment and venue of sex work. This high prevalence is consistent with findings from other studies, in which prevalence of workplace violence amongst FSW ranges between 32% and 75% (Deering et al., 2014). We also found that FSW rarely reported incidents of violence to the police, indicating a reluctance potentially tied to previous police mistreatment. Criminalising sex work leads FSW to practise in clandestine locations [START_REF] Muldoon | Policing the epidemic: High burden of workplace violence among female sex workers in conflict-affected northern Uganda[END_REF] such as their client's home, which-as demonstrated by our findings-exacerbates their risk of facing violence.

Environment as a correlate of violence

We observed that the location where FSW practised sex work was associated with violence.

Lusaka-based FSW had greater odds of facing violence than Ndola-based sex workers. This could be explained by the higher level of urbanisation in Lusaka-a city of ~3,000,000 inhabitants-compared to Ndola-a city of ~500,000 inhabitants (Central Statistical Office (Zambia), 2018). Rapid urbanisation in developing settings can exacerbate socioeconomic disparities such as poor housing and low income, which could then reinforce existing attitudes in favour of gender-based violence [START_REF] Patel | Rapid urbanization and the growing threat of violence and conflict: a 21st century crisis[END_REF]. However, rapid urbanisation is only one of several probable differences between Ndola and Lusaka. Our data suggest further exploration of regional or environmental differences in relation to client violence against FSW.

FSW who practised sex work in their clients' homes were highly likely to have experienced violence in our study. When in the client's home, an FSW may face a greater threat of violence due to the interplaying environmental aspects of control, concealment, and isolation [START_REF] Prior | Sex worker victimization, modes of working, and location in New South Wales, Australia: a geography of victimization[END_REF]. In his own home, a client might feel more in control and at liberty to be violent knowing that any such act would be concealed from the public. In the client' home, an FSW is isolated from any possible intervener in the event of violence. This calls for the creation of safe indoor workspaces, where FSW can negotiate safe sex with clients and avoid violence [START_REF] Krüsi | Negotiating Safety and Sexual Risk Reduction With Clients in Unsanctioned Safer Indoor Sex Work Environments: A Qualitative Study[END_REF]. In what is a hostile legal climate towards sex work, legal reform is required to ensure that such spaces do not engender structural violence.

Our finding that FSW who recruit clients from the street have increased odds of violence illustrates the danger of the outdoor environment for sex workers. Other studies corroborate that sex workers who solicit clients outdoors are more prone to violence than indoor sex workers are [START_REF] Katsulis | Predictors of Workplace Violence Among Female Sex Workers in Tijuana, Mexico[END_REF][START_REF] Miller | Violence and Coercion in Sri Lanka's Commercial Sex Industry: Intersections of Gender, Sexuality, Culture, and the Law[END_REF][START_REF] Panchal | Violence & Vulnerability: A Cross Sectional Study of Prevalence and Factors Associated with Sex-work Related Violence among Female Sex Workers[END_REF]. This further emphasises our call for policy makers to sanction safer, regulated indoor workspaces for FSW to solicit and service clients.

Sexual history as a correlate of violence

We found that violence was correlated with having a physically forced sexual debut. One third of FSW in our cohort were either verbally or physically coerced during their sexual initiation, which on average took place at age 15. This is in line with previous studies in sub-Saharan Africa, where the prevalence of forced sexual initiation among women varies between 5.3% and 46% [START_REF] Stockman | Forced Sexual Initiation, Sexual Intimate Partner Violence and HIV Risk in Women: A Global Review of the Literature[END_REF]. Similar to our findings, others have demonstrated an association between forced sexual debut and violence against FSW [START_REF] Argento | High Prevalence and Partner Correlates of Physical and Sexual Violence by Intimate Partners among Street and Off-Street Sex Workers[END_REF][START_REF] Erickson | Interpersonal and structural contexts of intimate partner violence among female sex workers in conflict-affected northern Uganda[END_REF]. Being sexually abused as a child can result in 'revictimisation' in adulthood [START_REF] Roodman | Revictimization rates and method variance: A metaanalysis[END_REF]. FSW with histories of child sexual abuse may thus perceive violence as inevitable and part of sex work or sexuality in general.

Existing violence prevention interventions for adolescent girls and young women must therefore target boys and young men to discourage physical and sexual abuse towards women.

Behaviour as a correlate of violence

We discovered that the odds of violence increased progressively for FSW who reported a higher volume of clients. This highlights the danger posed by frequent interactions with male clients who-as revealed in a Zambian survey-tend to have views in favour of violence against women [START_REF] Leclerc | Clients of Commercial Sex Workers in Zambia: Prevalence, Frequency, and Risk Factors[END_REF]. The primary motivation for FSW in our cohort entering the profession was financial. Hence, risk reduction interventions advocating for FSW to reduce the number of clients they see would be impractical. Our results do however support economic empowerment initiatives for FSW. As shown in Tanzania, economic empowerment can enable FSW to be more selective of their clients and thus surmount causes of vulnerability to HIV such as violence [START_REF] Mantsios | That's how we help each other': Community savings groups, economic empowerment and HIV risk among female sex workers in Iringa, Tanzania[END_REF]. It would also be prudent to develop educational interventions for male clients of FSW to redress the current gender norms that permit violence against women to persist.

Limitations

Our findings should be considered in light of certain limitations. Firstly, due to the crosssectional methodology of this study, we cannot infer causality from the results obtained.

Report of sexual violence was ever/never, and we were unable to determine the timing relative to the behavioural, sexual history, and environmental variables. Secondly, physical violence was self-reported and thus subject to social desirability bias, which was minimised by trained counsellors administering all surveys. Our study would have benefited from a more robust measure of violence; this could include sexual violence resulting in forced sex, and psychological violence in the form of threats and sexual coercion. Despite shedding light on client-perpetrated violence, our study lacked details on other perpetrators, i.e. intimate partners and police officers. Lastly, an examination of structural factors would have lent credence to our data. Indicators such as previous police harassment or arrest could explain the reluctance of FSW to report violence to the police.

Generalisability and implications

Our study makes a novel contribution to the limited literature on violence against Zambian FSW. To our knowledge, this is the first quantitative study to determine the prevalence and correlates of client-initiated violence against Zambian FSW. Broadly, our results point to the global problem of violence against women; this is a pertinent public health and human rights issue, which must be eliminated through cultural, policy and legal reform. The high (39%) prevalence of client-initiated violence in our cohort underscores the need to reform oppressive policies and customs that continue to endanger FSW. Much like Zambia, most countries in sub-Saharan Africa have gender norms and legal systems that compromise the safety of FSW [START_REF] Baleta | Lives on the line: sex work in sub-Saharan Africa[END_REF]. Our findings call for inclusive policies protecting the safety of FSW in sub-Saharan Africa, where gender-based violence is high and sex work remains stigmatised and criminalised.

Knowing that violence provokes poor sexual health outcomes, our findings support WHO recommendations to develop interventions targeting the dual burdens of violence and HIV (World Health Organization, 2012). The DREAMS programme (PEPFAR, USA) is one such integrated violence and HIV prevention intervention for adolescent girls and young women (aged 15-24); but, in Zambia, its coverage is limited to three cities (Lusaka, Ndola and Chingola), which represent less than 20% of the population (The President's Emergency Plan For AIDS Relief (PEPFAR), 2016). The scale up and broadening of such programmes to include sex workers could enhance the safety and sexual health of FSW. Epidemiological modelling in Kenya reveals that the elimination of violence alone could, within a decade, prevent 17% of incident HIV infections among sex workers [START_REF] Shannon | Global Epidemiology of HIV among Female Sex Workers: Influence of Structural Determinants[END_REF].

Conclusion

In our cohort of Zambian FSW, we found a high prevalence of client-initiated physical violence. The correlates of violence were client volume, history of forced sex, venue of recruitment and venue of sex work. This calls for behavioural risk-reduction interventions for FSW at the individual level and violence prevention programs for actual and potential male clients at the community level. Ultimately, legal and policy reform are required to create safer working environments for FSW. To help achieve this, the very frameworks that unjustly prosecute sex workers must instead direct their resources towards bringing violent clients to justice. Zulu, J., Ngwenya, Z., Silukena, M., Phiri, Z., & Chiseni, A. (2015). Integrating friendly sexual and reproductive health services for young female sex workers into the health system at district level in Zambia: perspectives of stakeholders. Health Systems and Policy Research, 2(1).

Table 1 .

 1 Descriptive characteristics of Zambian FSW in the study (n=419) comparing FSW who experienced client-initiated physical violence with those who did not

			Lifetime Violence
		Total Sample	Yes	No	X 2
		(N=419)	(n=164)	(n=255)
		n (%)	n (%)	n (%)	p-value
	Demographics
	A g e
	 18-20	127 (30)	52 (32)	75 (29)	0.497
	 21-25	142 (34)	50 (30)	92 (36)
	 26-45	150 (36)	62 (38)	78 (35)
	Education
	 None	35 (8)	17 (11)	18 (7)	0.204
	 Primary	202 (48)	71 (43)	131 (51)
	 Secondary or higher	182 (44)	76 (46)	106 (42)
	Reason for entering sex work
	 Financial	354 (84)	141 (86) 213 (83) 0.599
	 Nothing to do	11 (3)	4 (2)	7 (3)
	 Peer pressure	17 (4)	4 (2)	13 (5)
	 Other	37 (9)	15 (10)	22 (9)
	Sexual History
	Age at first sex
	 ≤13	78 (19)	24 (15)	54 (21)	0.288
	 14-15	134 (32)	59 (36)	75 (30)
	 16-19	174 (41)	69 (42)	105 (41)
	 20+	33 (8)	12 (7)	21 (8)
	Age at entry into sex work* n=2
	 ≤16	153 (36)	62 (38)	91 (36)	0.798
	 17-19	146 (35)	58 (35)	88 (34)
	 20+	120 (29)	44 (27)	76 (30)

This study was supported by the International AIDS Vaccine Initiative (IAVI) (https://www.iavi.org/) with the generous support of the American people through the United States Agency for International Development (USAID, https://www.usaid.gov/); National Institutes of Health (https://www.nih.gov/) grants (R01 MH66767, R01 HD40125, and R01 MH95503; R01 AI051231); the AIDS International Training and Research Program Fogarty International Center (D43 TW001042); and the Emory Center for AIDS Research (P30 AI050409).