Michel J G Weber

A BOUND FOR MEAN VALUES OF FOURIER TRANSFORMS

Keywords: 2010 AMS Mathematical Subject Classification: Primary: 60F15, 60G50 ; Secondary: 60F05 Fourier transform, mean value, convolution products, dilation, shift. ---------

We show that there exists a sequence {n k , k ≥ 1} growing at least geometrically such that for any finite non-negative measure ν such that ν ≥ 0, any T > 0,

Introduction

Let ν be a finite non-negative measure on R, ν(t) = R e itx ν(dx), then The first inequality is in turn true at any order: for any positive integer κ,

R sin T u/2 T u/2 2κ ν(du) ≤ 1 T κT -κT | ν(x)|dx. (1.
2)

The question whether the second inequality admits a similar extension arises naturally. We show the existence of a general form of that inequality in which appear constants growing fastly with κ.

Theorem 1.1. There exists a sequence {n k , k ≥ 1} growing at least geometrically such that for any finite non-negative measure ν such that ν ≥ 0, any T > 0, we have

2 n k T -2 n k T ν(x)dx ≪ ε T 2 2 (1+ε)n k R sin xT xT n 2 k ν(dx).
We don't know whether the constant 2 2 (1+ε)n k can be significantly weakened. The proof of is rather delicate. In order to prepare it, and also to provide the necessary hints concerning inequalities (1.1), (1.2), introduce some auxiliary functions 1 and indicate as well some related properties. Let K(t) = 1 -|t|) + , T > 0 and define K T (t) = K(t/T ) = 1 -|t|/T )χ {|t|≤T } . Then

K(u) = sin u/2 u/2 2 , K T (u) = 1 T sin T u/2 u/2 2 .
It is easy to check that K T (t) + K T (t + T ) + K T (t -T ) = 1, if |t| ≤ T . It follows that

χ {|t-H|≤T } ≤ K T (t -H) + K T (t -H + T ) + K T (t -H -T ).
( 1.3) This can be used to prove (1.1).

Since R K T (t -S) ν(t)dt = R e iSx K T (x)ν(dx), we deduce H+T H-T ν(t)dt ≤ R K T (t -H) + K T (t -H + T ) + K T (t -H -T ) ν(t)dt ≤ R K T (u) e iHu + e i(H-T )u + e i(H+T )u ν(du) = 1 T R sin T u/2 u/2 2 e iHu [1 + 2 cos T u]ν(du).
This immediately implies the second inequality in (1.1). Notice also by using Fubini's theorem, that for any reals S, γ, T reals, T > 0 and any integer κ > 0,

R sin T (u -γ)/2 T (u -γ)/2 2κ e iSu ν(du) = 1 T R e -iγ(y-S) K * κ y -S T ν(y)dy. (1.4) Letting κ = 1, γ = S = 0, gives 1 T R sin T u/2 u/2 2 ν(du) = R K T (y) ν(y)dy. As 0 ≤ K T (y) = 1 -|y|/T )χ {|y|≤T } ≤ χ {|y|≤T } , we deduce 1 T R sin T u/2 u/2 2 ν(du) ≤ R K T (y)| ν(y)|dy ≤ T -T | ν(y)|dy,
which yields the first inequality in (1.1).

As to (1.2), some properties of basic convolutions products are needed. Consider for A > 0 the elementary measures µ A with density

g µA (x) = χ {[-A,A]} (x). Let 0 < A ≤ B. Plainly g µA * µB (x) = g µA * g µB (x) = 2A 1 - |x| A + B • χ {[-A-B,A+B]} (x). (1.5) Indeed, g µA * µB (x) = R g µA (x -y)g µB (y)dy = B -B χ {[x-A,x+A]} (y)dy, and this is equal to λ [-B, B] ∩ [x -A, x + A] = B ∧ (x + A) -(x -A) ∨ (-B) χ {[-A-B,A+B]} (x).
In particular, introducing the function g

(x) = χ [-1 2 , 1 2 ] (x), we have K(t) = g * g(t). More generally, Lemma 1.2. Let 0 < A 1 ≤ A 2 ≤ . . . ≤ A J and µ = µ A1 * µ A2 * . . . * µ AJ . Then µ has density g satisfying 0 ≤ g(x) ≤ G J • χ {[-(A1+A2+...+AJ ),A1+A2+...+AJ ]} (x),
where

G J = 2 J A 1 • (A 1 + A 2 ) ∧ A 3 • (A 1 + A 2 + A 3 ) ∧ A 4 . . . (A 1 + . . . + A J-1 ) ∧ A J .
Proof. We prove it by induction. By (1.5), for every real x

0 ≤ g µA 1 * µA 2 (x) ≤ 2A 1 • χ {[-A1-A2,A1+A2]} (x) = 2A 1 • g µA 1 +A 2 (x).
The case J = 2 is proved. Now for J = 3, by what preceeeds

g µA 1 * µA 2 * µA 3 (x) = A3 -A3 g µA 1 * µA 2 (x -y)dy ≤ 2A 1 A3 -A3 g µA 1 +A 2 (x -y)dy = 2A 1 g µA 1 +A 2 * µA 3 ≤ 2A 1 2(A 3 ∧ A 1 + A 2 ) • χ {[-A1-A2-A3,A1+A2+A3]} (x).
The general case follows by iterating the same argument.

In particular, for any positive J,

0 ≤ K * J (x) ≤ χ {[-J,J]} (x).
(1.6) Indeed, apply Lemma 1.2 with A j ≡ 1/2. We get

0 ≤ K * J (x) = g * 2J (x) ≤ G 2J • χ {[-J,J]} (x),
and

G 2J = 2 2J • 2 -2J = 1. Inequality (1.2
) is yet a direct consequence of (1.4) and (1.6). Now, we pass to the preparation of the proof of Theorem 1.1, and begin to explain how we shall proceed. By using (1.3) with H = 0, T = 1/2, we get g(x) ≤ g * 2 (2x) + g * 2 (2x + 1) + g * 2 (2x -1).

(1.7)

An important intermediate step towards the proof of Theorem 1.1 will consist to generalizing that inequality. Our approach can be described as follows. As g * 2 (2v) = R g(2v -y)g(y)dy, (1.7) can be used to bound the integration term g(y). And by next reporting this into (1.7), it follows that g(x) can also be bounded by a sum of terms of type g * 3 . Call E this operation. By iterating E, we similarly obtain variant forms of (1.3), involving higher convolution powers of g. The study of the iterated action of E, as well as the order of the constants generated is made in the next section. The action of E will be first described as the combination of two elementary transforms acting alternatively.

Stacks and shifts

We first introduce some operators and related auxiliary results, as well as the necessary notation. Given f : R → R and a > 0, let

T a f (x) = f ( x a ) be the dilation of f by a -1 . Plainly T a T b = T ab . Notice also that T a (h * f ) = 1 a T a h * T a f, h ∈ L 1 (R), f ∈ L ∞ (R). (2.1) Indeed T a (h * f )(u) = R h( u a -x)f (x)dx = R T a h(u -ax)T a f (ax)dx = 1 a R T a h(u -v)T a f (v)dv = 1 a T a h * T a f (u).
More generally

T a (h 1 * . . . * h n ) = 1 a n-1 T a (h 1 ) * . . . * T a (h n ), h 1 , . . . , h n ∈ L ∞ (R). (2.2)
Introduce also the sequence of g-dilations

g k = T 2 -k g, k = 1, 2, . . . . Now let I be a finite subset of R. It will be convenient to denote Σ[f (x) : I] = ρ∈I f (x + ρ). (2.3) We have Σ[f (bx) : I] = Σ[T 1 b f (x) : 1 b I]. (2.4) And Σ[f : I](t) = f (t) ρ∈I e -itρ . (2.5)
The linear operator f → Σ[f : I] on L 1 (R) commutes with the convolution operation: (2.7) Let j 0 < j 1 . . . < j k be a finite set of positive integers, which we denote J. Let C = {c j , j ∈ J} be some other set of positive integers, not necessarily distinct. We identify (J, C) with U := {(j, c j ), j ∈ J}, and put

f * Σ[h : I] = Σ[f * h : I]. (2.6) Further Σ[f : I] ≥ 0 if f ≥ 0. We
J * = J if c j0 > 1 J\{j 0 } if c j0 = 1.
(2.8)

Define the transform J → J 1 as follows

J 1 = D(J) := J * ∪ (j 0 + J). (2.9) Next define C → C 1 by putting C 1 = T (C) := {c 1 j , j ∈ J 1 }, (2.10)
where

c 1 j =          c j + c j-j0 if j ∈ J * ∩ (j 0 + J) c j-j0 if j ∈ (J * ) c ∩ (j 0 + J) c j if j ∈ J * ∩ (j 0 + J) c and j > j 0 c j0-1 if j 0 ∈ J * .
(2.11)

Similarly we identify (J 1 , C 1 ) with U 1 := {(j, c 1 j ), j ∈ J 1 } The successive transforms (J, C) → (J 1 , C 1 ) → (J 2 , C 2 ) → . . . turn up to describe the iterated of E, and may be compared to the action of superposing shifted functions. We start with J = {1}, C = {2} corresponding to the basic set

U = {(1, 2)}.
It is easy to check that the iterated transforms of U progressively generate the sequence of sets [START_REF] Mitrinović | Analytic inequalities[END_REF][START_REF] Mitrinović | Analytic inequalities[END_REF], [START_REF] Weber | [END_REF][START_REF] Weber | [END_REF] (2, 3), (3, 2) (2, 2), (3, 2), (4, 3), (5, 2) (2, 1), (3, 2), (4, 5), (5, 4), (6, 3), (7, 2) (3, 2), (4, 6), (5, 6), (6, 8), (7, 6), (8, 3), (9, 2) (3, 1), (4, 6), (5, 6), (6, 10), (7, 12), (8, 9), (9, 10), (10, 6), (11, 3), (12, 2) . . .

At the m-th step, the set J m is an interval of integers {a m , . . . , b m } with a m → ∞ slowly, whereas b m → ∞ very rapidly. More precisely, let for k = 1, 2, . . .

r k = max m≥1 c m k .
Then r 1 = 2, r 2 = 3, r 3 = 2, r 4 = 6, . . . etc. And define

R k = r 1 + . . . + r k , ζ k = 1 + r 1 + 2r 2 + . . . + kr k . (2.12) Let R k-1 < m ≤ R k .
At step m, J m is realized by first shifting J m-1 on the right from a length k, next taking union with J * m-1 and in turn

J m = k, k + 1, . . . , ζ k-1 + (m -R k-1 )k , if m < R k , whereas J R k = k + 1, . . . , ζ k . Write m = R k-1 + h, 1 ≤ h ≤ r k . Then we have the relations c m j = c R k-1 j-k + [r k + . . . + (r k -h + 1)] 2k ≤ j ≤ ζ k-1 + (m -R k-1 )k, c R k-1 j k ≤ j < 2k. (2.13) After the steps R k-1 + 1, R k-1 + 2, . . . , R k , the function h → c R k-1 +h n will have increased from r k + (r k -1) + . . . + 2 + 1 = r k (r k + 1) 2 for all n ∈ 2k, . . . , ζ k-1 . It follows that min n∈{2k,...,ζ k-1 } r n ≥ r 2 k 2 .
(2.14) Therefore r 2k ≥ r 2 k /2. This being true for all k, yields by iteration

r 2 j ≥ 1 2 (r 2 j-1 ) 2 ≥ 1 2 1 2 2 (r 2 j-2 ) 2 2 ≥ . . . ≥ 1 2 1+2+...+2 H-1 (r 2 j-H ) 2 H = r 2 j-H 2 2 H
.

We have r 2 = 3. Thus

r 2 j ≥ 3 2 2 j-1 , j = 1, 2, . . . . (2.15)
We shall deduce from this and (2.14) that r k grows at least geometrically. Let n and let j be such that 2 j+1 ≤ n < 2 j+2 . Apply (2.14) with k = 2 j . As n ≥ 2k, we have

r n ≥ r 2 2 j /2 once 2 j+2 ≤ ζ 2 j -1 . But ζ 2 j -1 ≥ ζ 2 j-1 = 1 + r 1 + 2r 2 + . . . + 2 j-1 r 2 j-1 ≥ 2 j-1 3 2 2 j-2 ≫ 2 j+2 .
Thereby, for j large

r n ≥ 1 2 r 2 2 j ≥ 1 2 3 2 2 j-1 = 1 2 3 2 2 j+2 8 ≥ 1 2 3 2 n 8 = 1 2 e ( 1 8 log 3 2 )n .
Consequently, there is a numerical constant ρ > 1, such that for all n ≥ 1, we have

r n ≥ ρ n . (2.16) Let j m := #{J m }. Since j m = ζ k-1 + (m -R k-1 -1)k if R k-1 < m ≤ R k , we have R k-1 <m≤R k j m = R k-1 <m≤R k (ζ k-1 + (m -R k-1 -1)k) = r k ζ k-1 + k r k -1 u=1 u.
We thus notice for later use that

R k-1 <m≤R k j m = r k ζ k-1 + k r k (r k + 1) 2 .
(2.17) Let * j f j denotes the convolution product of f j 's. Finally we put

I = -1 2 , 0, 1 2 .
Our next result generalizes inequality (1.3) to arbitrary convolution powers of g.

Proposition 2.1. Let k ≥ 1 and R k-1 < m ≤ R k . Then g(x) ≤ C m Σ * (j,cj)∈Um g * cj j (x) : I m ,
where I m , C m are defined by the recurrence relations:

I 0 = -1 2 , 0, 1 2 , C 0 = 2 and I m = I m-1 + r k I m-1 , C m = 2 k(jm-1-1) C 2 m-1 .
Proof. We use repetitively the relation (see (2.1))

T 1 2 (h * f ) = 2 T 1 2 h * T 1 2 f, f ∈ L ∞ (R), h ∈ L 1 (R). By (1.7), g(x) ≤ 2 g * 2 1 (x) + g * 2 1 (x + 1 2 ) + g * 2 1 (x - 1 2 ) = C 0 Σ g * 2 1 (x) : I = Σ * (j,cj )∈U g * cj j (x) : I . (2.18)
Now we apply E. We begin with the "stack" of 1's of height r 1 = 2. At first

g * 2 1 (x) = R g 1 (x -y)g 1 (y)dy = R g 1 (x -y)g(2y)dy ≤ C 0 R g 1 (x -y)Σ g * 2 1 (2y) : I dy.
But by (2.4), next (2.1)

Σ g * 2 1 (2y) : I = Σ T 1 2 (g * 2 1
)(y) :

1 2 I = Σ 2(T 1 2 g 1 ) * 2 (y) : 1 2 I = 2Σ g * 2 2 (y) : 1 2 I . Therefore g * 2 1 (x) ≤ 2C 0 R g 1 (x -y)Σ g * 2 2 (y) : 1 2 I dy = 2C 0 Σ g 1 * g * 2 2 (x) : 1 2 I .
By reporting in (2.18), we obtain

g(x) ≤ C 0 (2C 0 )Σ Σ g 1 * g * 2 2 (x) : 1 2 I : I = C 1 Σ g 1 * g * 2 2 (x) : I 1 = C 1 Σ * (j,cj)∈U1 g * cj j (x) : I 1 . ( 2

.19)

And C 1 = 8. We now apply E once again, and bound the generic product g 1 * g * 2 2 (x) by applying (2.19)

to g 1 . Concretely R g * 2 2 (x -y)g 1 (y)dy = R g * 2 2 (x -y)g(2y)dy ≤ C 1 R g * 2 2 (x -y)Σ g 1 * g * 2 2 (2y) : I 1 dy = 2 3-1 C 1 R g * 2 2 (x -y)Σ g 2 * g * 2 3 (y) : 1 2 I 1 dy = 2 2 C 1 Σ g * 3 2 * g * 2 3 (x) : 1 2 I 1 .
By reporting in (2.19), we obtain

g(x) ≤ C 1 (2 2 C 1 )Σ Σ g * 3 2 * g * 2 3 (x) : 1 2 I 1 : I 1 = C 2 Σ g * 3 2 * g * 2 3 (x) : I 2 = C 2 Σ * (j,cj )∈U2 g * cj j (x) : I 2 . (2.20)
And C 2 = 256. For the next E-iteration, as we have exhausted the stack of 1's, we now use the stack of 2's of height r 2 = 3. We bound the new the generic product g * 3 2 * g * 2 3 (x) by applying (2.20) to g 2 (x) as follows:

R g * 2 2 * g * 2 3 (x -y)g 2 (y)dy = R g * 2 2 * g * 2 3 (x -y)g(2 2 y)dy ≤ C 2 R g * 2 2 * g * 2 3 (x -y)Σ g * 3 2 * g * 2 3 (4y) : I 3 dy = 2 2(3+2-1) C 2 R g * 2 2 * g * 2 3 (x -y)Σ g * 3 4 * g * 2 5 (y) : 1 4 I 2 dy = 2 8 C 2 Σ g * 2 2 * g * 2 3 * g * 3 4 * g * 2 5 (x) : 1 4 I 2 .
By reporting in (2.20), we obtain

g(x) ≤ 2 8 C 2 2 Σ Σ g * 2 2 * g * 2 3 * g * 3 4 * g * 2 5 (x) : 1 4 I 2 : I 2 = C 3 Σ g * 2 2 * g * 2 3 * g * 3 4 * g * 2 5 (x) : I 3 , (2.21)
with C 3 = 16777216. And so on.

To simplify, let k ≥ 1 and R k-1 < m ≤ R k . At step m, we play with the stack of k's of height r k and apply the bound previously obtained to the least dilation of g in the generic product G = * (j,cj )∈Um-1 g * cj j (x) from the previous step. The dilation factor being 2 k , the bound of g k (x) thereby produces the new terms

T 2 -k (G)(x) = * (j,cj)∈Um-1 g * cj j+k (x).
Hence by (2.2), after integration, a constant factor 2 k(jm-1-1) C m-1 . Next we report the bound obtained for the generic products in the inequality from the preceding step. This is exactly what describes transform D. This generates a new constant factor C m-1 . Together with the preceding constant factor, this gives the constant 2 k(jm-1-1) C 2 m-1 = C m . The rule concerning constants C m being the same at each step inside the block ]R k-1 , R k ], we have the recurrence relation

C m = 2 k(jm-1-1) C 2 m-1 . (2.22) And the transform c m-1 j → c m j is described by T . Let k ≥ 1. Put γ k = j∈JR k c R k j , d k = j∈JR k jc R k j .
(2.23)

We shall now deduce the following estimate.

Proposition 2.2. Let ν be a finite measure such that ν ≥ 0. Then for any W > 0

1 2W W -W ν(t)dt ≤ C R k 2 -d k +1 R (j,cj )∈UR k sin( 2W x 2 j ) 2W x 2 j cj ρ∈2W IR k e -iρx ν(dx).
Proof. Recall that J R k = k + 1, . . . , ζ k . Further, by (2.12)

γ k ≥ r 2 k 2 (ζ k-1 -2k) = r 2 k 2 ζ k-1 (1 - 2k ζ k-1 ) ≥ ( 1 -ε 2 )r 2 k ζ k-1 , (2.24) once k ≥ k ε . Similarly d k ≥ r 2 k 2 j∈{2k,...,ζ k-1 } j ≥ r 2 k 4 (ζ 2 k-1 -4k 2 ) ≥ ( 1 -ε 4 )r 2 k ζ 2 k-1 (2.25) for k large enough. By Proposition 2.1, with m = R k g(t) ≤ C R k Σ * (j,cj)∈UR k g * cj j (t) : I R k .
Let W > 0. Then by (2.4), next (2.2)

χ [-W,W ] (t) = g( t 2W ) ≤ C R k Σ * (j,cj )∈UR k g * cj j ( t 2W ) : I R k = C R k Σ T 2W * (j,cj)∈UR k g * cj j (t) : 2W I R k = C R k (2W ) γ k -1 Σ * (j,cj)∈UR k (T 2W g j ) * cj (t) : 2W I R k . (2.26) By (2.5) Σ * (j,cj )∈UR k (T 2W g j ) * cj : 2W I R k (x) = (j,cj)∈UR k T 2W g j (x) cj ρ∈2W IR k e -iρx . (2.27) But T 2W g j (x) = R e ixu g( 2 j u 2W )du = 2W 2 j R e i2W 2 j xv g(v)dv = 2W 2 j g( 2W x 2 j ). Hence Σ * (j,cj )∈UR k (T 2W g j ) * cj : 2W I R k (x) = (2W ) γ k 2 -d k (j,cj )∈UR k g 2W x 2 j cj ρ∈2W IR k e -iρx .
(2.28)

And by the Parseval relation

1 2W W -W ν(t)dt ≤ C R k (2W ) γ k R Σ * (j,cj )∈UR k (T 2W g j ) * cj (t) : 2W I R k ν(t)dt = C R k (2W ) γ k (2W ) γ k 2 -d k R (j,cj)∈UR k g( 2W x 2 j ) cj ρ∈2W IR k e -iρx ν(dx) = C R k 2 -d k +1 R (j,cj )∈UR k sin( 2W x 2 j ) 2W x 2 j cj ρ∈2W IR k e -iρx ν(dx).
(2.29)

3. Proof of Theorem 1.1 By assumption ν ≥ 0. Choose W = 2 ζ k T . Then sin( 2W x 2 j ) 2W x 2 j = sin(2 ζ k +1-j xT ) 2 ζ k +1-j xT But we have that sin n k=1 x k ≤ n k=1 sin x k , (3.1)
if 0 < x k < π and n > 1, see [START_REF] Mitrinović | Analytic inequalities[END_REF] p.236. From this easily follows that | sin nx| ≤ n| sin x| for any real x and any integer n. Indeed, write x = x ′ +kπ with 0 < x ′ < π.

Then | sin nx| = | sin(nx ′ + nkϕ)| = | sin nx ′ | ≤ n| sin nx ′ | = n| sin nx|. Consequently sin 2 ζ k +1-j xT 2 ζ k +1-j xT ≤ sin xT xT . (3.2)
By reporting and since #{I

R k } = 3 R k we get 1 2.2 ζ k T 2 ζ k T -2 ζ k T ν(dt) ≤ C R k 3 R k 2 -d k R sin xT xT γ k ν(dx).
(3.3)

And by using estimates (2.24), (2.25)

1 2.2 ζ k T 2 ζ k T -2 ζ k T ν(dt) ≤ C R k 3 R k 2 -1 5 r 2 k ζ 2 k-1 R sin xT xT 1 3 r 2 k ζ k-1
ν(dx).

(3.4)

We now estimate C R k . By iterating inside the block of integers ]R k-1 , R k ] the recurrence relation C m = 2 k(jm-1-1) C 2 m-1 obtained in Proposition 2.1, we obtain C R k = 2 k{(jR k-1 -1)+...+(jR k -1)} C 2 r k R k-1 . According to (2.17 As ζ k = 1 + r 1 + 2r 2 + . . . + kr k , it follows that

kr k (ζ k-1 -1) + k 2 r k (r k + 1) 2 ≤ kr k ζ k-1 + k 2 r 2 k ≤ ζ k ζ k-1 + ζ 2 k ≤ 2ζ 2 k . Thus C R k ≤ 2 2ζ 2 k C 2 r k R k-1 .
(3.5) By successively iterating this, and since C R1 = 2, we get

C R k ≤ 2 2{ζ 2 k +ζ 2 k-1 2 r k +ζ 2 k-2 (2 r k +2 r k-1 )+...+ζ 2
2 (2 r k +...+2 r 3 ))} 2 2 r k +2 r k-1 +...+2 r 2 ≤ 2 2.2 r k kζ 2 k .

But r k ≥ ρ k by (2.16), so that

R k ≤ ζ k = 1 + r 1 + 2r 2 + . . . + kr k ≪ ε 2 εr k .
Hence also C R k ≤ 2 2 (1+ε)r k .

(3.6) Finally, 1 T (3.8)

2 ζ k T -2 ζ k T ν(dt) ≪ ε 2 2 (1+ε)r k 2 ζ k -1 5 r 2 k ζ 2 k-1 R sin xT xT

  use the standard arithmetical set notation: λI = {λρ : ρ ∈ I} and if I, J are two finite subsets, I + J = {ρ + η : ρ ∈ I, η ∈ J}, repetitions are counted. This is relevant since Σ Σ[f (x) : I] : J = Σ[f (x) : I + J].

  ), we havek{(j R k-1 -1) + . . . + (j R k -1)} = k R k-1 <m≤R k (j m -1) = kr k (ζ k-1 -1) + k 2 r k (r k + 1)2 .


Michel Weber

A Sharp Estimate for Divisors of Bernoulli Sums

Let Sn = ε1 + . . . + εn, where εi are i.

]. We show that sup 2≤d≤n P d|Sn -E(n, d) = O log 5/2 n n 3/2 , where E(n, d) verifies c1β(n, d) ≤ E(n, d) ≤ c2β(n, d) and c1, c2 are numerical constants.

Main result

Let {ε i , i ≥ 1} denote a Bernoulli sequence defined on a joint probability space ( Ω, Ã, P), with partial sums S n = ε 1 + . . . + ε n . Consider the Theta function

Θ(d, m) = ℓ∈Z e imπ ℓ d -mπ 2 ℓ 2 2d 2 .
The improvment of the following result, which is Theorem II in [START_REF] Weber | Small divisors of Bernoulli sums[END_REF], is the main purpose of this work.

Proposition 1 We have the following uniform estimate:

sup 2≤d≤n P d|S n - Θ(d, n) d = O log 5/2 n n 3/2 . ( 1
)
This estimate is sharp already when d < (Bn/ log n) 1/2 , otherwise

P d|S n - 1 d ≤ C log 5/2 n n 3/2 + 1 d e -nπ 2 2d 2 if d ≤ n 1/2 , C n 1/2 if n 1/2 ≤ d ≤ n. ( 2
)
And this is no longer efficient when d ≫ √ n. The purpose of this Note is to remedy this by showing the existence of an extra corrective exponential factor in that case. Introduce a notation. Let n ≥ d ≥ 2 be integers and denote by r d (n) the least residue of n modulo 2d: n ≡ r mod(2d) and 0 ≤ r < 2d. Let also denote rd (n) = 2dr d (n).

Theorem 2 We have

sup 2≤d≤n P d|S n -E(n, d) = O log 5/2 n n 3/2 .
where E(n, d) satisfies

1 2 √ 2π ≤ E(n, d) max 1 2d , 1 √ n e -r d (n) 2 2n + e -rd (n) 2 2n ≤ 32 √ 2π .
This exponential factor is effective when min(r d (n), rd (n)) ≫ √ n. Its importance is easily seen through the following example.

Let 0 < c < 1 and let 1 ≤ ϕ 1 (n) ≤ cϕ 2 (n) be non-decreasing. Suppose d is such that 2d ≥ √ nϕ 2 (n) with r d (n) large so that √ nϕ 1 (n) ≤ r d (n) ≤ c √ nϕ 2 (n). Then rd (n) ≥ (1 -c) √ nϕ 2 (n) and so E(n, d) ≤ 32 √ 2πn e - ϕ 2 1 (n) 2 + e - (1-c) 2 ϕ 2 2 (n) 2 . Let 0 < A 1 ≤ A 2 . By taking ϕ i (n) = √ 2A i log n, i = 1, 2, we get E(n, d) ≤ C max n -1/2-A1 , n -1/2-(1-c) 2 A2 ≪ n -1/2 .
Thus we get a much better upper bound than in [START_REF] Weber | Small divisors of Bernoulli sums[END_REF]. The proof uses estimates for Theta functions, which are provided in the next Section.

2 Theta Function Estimates

Let E(n, d) := Θ(d,n) d . By the Poisson summation formula ℓ∈Z e -(ℓ+δ) 2 πx -1 = x 1/2 ℓ∈Z e 2iπℓδ-ℓ 2 πx ,
where x is any real and 0 ≤ δ ≤ 1, we get with the choices x = πn/(2d 2 ),

δ = n/(2d) E(n, d) = 2 πn h∈Z e -2({ n 2d }+h) 2 d 2 n . (3)
Let a > 0, 0 ≤ µ ≤ 1 and write μ := 1µ. We begin with elementary estimates of

S(µ, a) := h∈Z e -a(µ+h) 2 = e -aµ 2 + e -aμ 2 + ∞ h=1 e -a(µ+h) 2 + ∞ h=1 e -a(h+μ) 2 . Lemma 3 Define for 0 ≤ µ ≤ 1 and a > 0, ϕ(µ, a) = 1 √ 2a+2aµ . Then (ϕ(µ, a) -1)e -aµ 2 ≤ ∞ h=1 e -a(µ+h) 2 ≤ 2ϕ(µ, a)e -aµ 2 . Proof. Consider Mill's ratio R(x) = e x 2 /2 ∞
x e -t 2 /2 dt. Then ([1] section 2.26)

1 1 + x ≤ 2 √ x 2 + 4 + x ≤ R(x) ≤ 2 x 2 + 8/π + x ≤ 2 1 + x , x ≥ 0. First ∞ 0 e -a(µ+x) 2 dx -e -aµ 2 ≤ ∞ h=1 e -a(µ+h) 2 ≤ ∞ 0 e -a(µ+x) 2 dx. But ∞ 0 e -a(µ+x) 2 dx = e -µ 2 a √ 2a R(µ √ 2a) and 1 1 + µ √ 2a ≤ R(µ √ 2a) ≤ 2 1 + µ √ 2a . Thus 1 √ 2a + 2aµ e -aµ 2 ≤ ∞ 0 e -a(µ+x) 2 dx ≤ 2 √ 2a + 2aµ e -aµ 2 . Hence (ϕ(µ, a) -1)e -aµ 2 ≤ ∞ h=1 e -a(µ+h) 2 ≤ 2ϕ(µ, a)e -aµ 2 ,
as claimed.

Corollary 4 Put ψ(µ, a) := 1 + ϕ(µ, a) e -aµ 2 . Then for every 0 ≤ µ 1 and

a > 0 1 2 ≤ S(µ, a) ψ(µ, a) + ψ(μ, a) ≤ 2.

Proof. At first by the previous Lemma

A := e -aµ 2 + ∞ h=1 e -a(µ+h) 2 ≤ e -aµ 2 1 + 2ϕ(µ, a) .

Next A ≥ e -aµ 2 + 1 2 ∞ h=1 e -a(µ+h) 2 ≥ e -aµ 2 + 1 2 ϕ(µ, a)-1 e -aµ 2 = 1 2 1+ϕ(µ, a) e -aµ 2 . Thereby 1/2 ≤ A ψ(µ,a) ≤ 2. Operating similarly with Ā = e -aμ 2 + ∞ h=1 e -a(μ+h) 2 leads to 1 2 ≤ S(µ, a) ψ(µ, a) + ψ(μ, a) ≤ 2.
Notice that ϕ(0, a) = 1/ √ 2a and

1 2 1 + 1 √ 2a ≤ S(0, a) = 1 + 2 ∞ h=1 e -ah 2 ≤ 4 1 + 1 √ 2a . (4)
We now need an extra Lemma.

Lemma 5 Let n = 2dK + r with 1 ≤ r ≤ 2d. Then

1 2 max 1 2d , 1 √ n e -r 2 2n ≤ ψ( r 2d , 2d 2 n ) √ n ≤ 2 max 1 2d , 1 √ n e -r 2 2n .
Proof. We have

ψ( r 2d , 2d 2 n ) = 1 + √ n 2d 1 1 + r √ n e -r 2 2n .
We consider three cases.

Case a. 2d ≤ √ n. Then r √ n < 2d
√ n ≤ 1, and so

√ n 4d e -r 2 2n ≤ ψ( r 2d , 2d 2 n ) ≤ √ n d e -r 2 2n , which implies 1 2 max 1 2d , 1 √ n e -r 2 2n = e -r 2 2n 4d ≤ ψ( r 2d , 2d 2 n ) √ n ≤ e -r 2 2n d = 2 max 1 2d , 1 √ n e -r 2 2n . Case b. 2d ≥ √ n and r ≤ √ n. Here we have e -r 2 2n ≤ ψ( r 2d , 2d 2 n ) ≤ 2e -r 2 2n , which implies max 1 2d , 1 √ n e -r 2 2n = e -r 2 2n √ n ≤ ψ( r 2d , 2d 2 n ) √ n ≤ 2e -r 2 2n √ n = 2 max 1 2d , 1 √ n e -r 2 2n . Case c. 2d ≥ √ n and r ≥ √ n.
The exponential factor e -r 2 2n may this time play a role (if r ≫ √ n), and we have e

-r 2 2n ≤ ψ( r 2d , 2d 2 n ) ≤ 3 2 e -r 2 2n which implies max 1 2d , 1 √ n e -r 2 2n = e -r 2 2n √ n ≤ ψ( r 2d , 2d 2 n ) √ n ≤ 3e -r 2 2n 2 √ n = 3 2 max 1 2d , 1 √ n e -r 2 2n
.

Summarizing cases a) to c), we have that

1 2 max 1 2d , 1 √ n e -r 2 2n ≤ ψ( r 2d , 2d 2 n ) √ n ≤ 2 max 1 2d , 1 √ n e -r 2 2n .

Proof

A first case is simple.

Case I. 2d|n. We have E(n, d) = 2 πn S(0, 2d 2 n ). But by (4)

1 2 max 1, √ n 2d ≤ 1 2 1 + √ n 2d ≤ S(0, 2d 2 n ) ≤ 4 1 + √ n 2d ≤ 8 max 1, √ n 2d . Hence 1 √ 2π max 1 2d , 1 √ n ≤ E(n, d) ≤ 16 √ 2π max 1 2d , 1 √ n . (5)
Case II. Now if 2d |n, write n = 2dK + ρ with 0 < ρ < 2d. In our setting a

= 2d 2 n , µ = { n 2d } = ρ 2d and by (3), E(n, d) = 2/πn S({ n 2d }, 2d 2 n ). Applying Lemma 5 with r = ρ gives 1 2 max 1 2d , 1 √ n e -ρ 2 2n ≤ ψ( ρ 2d , 2d 2 n ) √ n ≤ 2 max 1 2d , 1 √ n e -ρ 2 2n . (6)
As to ψ(μ, 2d 2 n ), we have μ = 2d-ρ 2d := ρ 2d and 0 < ρ < 2d. Applying Lemma 5 with r = ρ gives

1 2 max 1 2d , 1 √ n e - ρ2 2n ≤ ψ( ρ 2d , 2d 2 n ) √ n ≤ 2


Sabri Hamana
email: sabri.hamana@mines-stetienne.fr

Vincent Augusto
email: augusto@mines-stetienne.fr

Xiaolan Xie
email: xie@mines-stetienne.fr

Modelling Interactions between Health Institutions in the Context of Patient Care Pathway

Keywords: patient care pathway, hospital, City health facilities, information systems, interactions, generic modelling framework

Health systems around the world have been continuously subject to many questions about their performance. In order to improve the functioning of such systems, Health Information Systems (HIS) and Technologies (HIT) are deployed to support patient care pathways and ensure information exchange between health structures, actors of patient care. The aim of this paper is to propose a generic modelling framework for describing the exchange of information between health institutions in the context of patient's care, concerning a given chronic pathology. The purpose is to facilitate building of flow models starting from the proposed modelling framework, those models will be used to support quantitative evaluation through discrete event simulation, useful to evaluate the impact of communication between health institutions on patient care pathway.

Introduction

Health systems around the world are in crisis because of the increase of care expenditures whereas resources become limited [START_REF] Bourret | Partager l'information pour dépasser les frontières institutionnelles et professionnelles ou le défi des réseaux de santé dans le système de santé français[END_REF]. The French health system is constrained in the same way and faces since many years an increase mainly related to care spending which can be explained, in part, by the increase in care demand due to the population aging and also the increase of the number of people with chronic diseases. On the other hand, the increase in health spending is explained by the many frontiers that separate the French health system and generate costs: the barriers, now considered as non-quality, could explain up to 15% unjustified costs [START_REF] Bourret | Partager l'information pour dépasser les frontières institutionnelles et professionnelles ou le défi des réseaux de santé dans le système de santé français[END_REF], the main partitioning being between hospitals and private practitioners.

To tackle this problem, it is important to reorganize the health system around the patient's care pathway [START_REF]Stratégie nationale de santé et enjeux régionaux d'une médecine de parcours; Comment améliorer le service rendu au patient[END_REF]. Given that the hospital represents itself a care pathway, organized and coordinated by the hospital information system, it seems judicious to export that model outside the hospital walls, by implementing e-health platforms and the interfaces between such platforms and all the information systems of health institutions within a territory. The goal consists in having a health information system which is computerized and integrated, enabling the continuity of care and the traceability in patients' pathways. Such approach is motivated by the fact that organizational problems related to patient care (delays between health-care stays, ruptures in patient pathway…) are primarily related to a lack of communication between health institutions, which lead to a lack of coordination between the different health actors involved in the patient's care [START_REF] Caillavet-Bachellez | Le parcours de soins du patient : un dispositif fédérateur pour les Communautés Hospitalières de Territoire (CHT)[END_REF].

This work is a part of a research project entitled e-SIS (French acronym for "Assessment in Health Information Systems"), project funded by the French health ministry (DGOS) as part of its research program called PREPS (French acronym for "Research program on performance of health-care systems"). The project aims to assess the impact of information systems on creating value in health facilities.

As regards the quantitative evaluation of HIS, there is no study that takes into account, at the same time, human, organizational and technological aspects of an IS and their impacts on values' creation within health institutions [START_REF] Yusof | Investigating evaluation frameworks for health information systems[END_REF]. Whereas for the modelling of HIS, some authors have proposed flow models which were described using graphical formalisms such as UML [START_REF] Spyrou | Extending UML activity diagrams for workflow modelling with clinical documents in regional health information systems[END_REF] and 3LGM 2 [START_REF] Winter | Modeling hospital information systems (part 1): the revised three-layer graph-based meta model 3LGM2[END_REF], except that these models describe only the flow of electronic information among computer systems within an IS, neglecting thus the human and organizational aspects. Other authors, for their part, have proposed models to exclusively describe patients' flows through their care pathway, using formalisms such as BPMN, SADT or also Petri Nets [7] [8].Therefore, there has not been, to our knowledge, a modelling framework that jointly describe the flows of patients and information within a one and only model representing a health institution or territory.

Thus, before achieving such evaluation, a modelling work of cancer care system should be performed upstream. For that reason, we decided to implement a modelling framework based on the process approach. Then starting from the proposed modelling framework we belt a flow model which will support the quantitative analysis (through discrete event simulation), the ultimate objective being to evaluate the impact of information exchanges between a hospital and a city health facilities on patient cancer pathway.

This paper is organized as follows: position of the problem is presented in Section 2. A generic modelling framework for describing the management system of information exchanged between health institutions within the context of patient pathway is described in Section 3. Finally, conclusions and perspectives are given in Section 4.

Position of the Problem

Anyone who is suffering from a chronic disease, such as cancer, follows a double care pathway: (i) In-hospital, within a hospital centre or a specialized centre for cancer care, and (ii) out-hospital, within city health facilities (General Practitioners (GP) offices, biological analysis laboratories, rehabilitation care facilities, hospital at home structures...). The role of hospitals is to provide a highly specialized diagnosis and treatment, which will be supported by an out-hospital care, provided by city health facilities.

On the other hand, patient's care at a health facility (hospital or city facility) requires external medical information provided by other health facilities; patient's care at that facility will also create, by the same occasion, medical information that will be required as a result of patient care in other health facilities. It is noteworthy that the exchange of information between institutions is done through one or more communication interfaces (e.g. postal mail, secure electronic messaging, shared patient record, etc.), each interface being characterized by a cost, related to resources used to transmit information, as well as an information delivery delay.

Given that the actual literature does not provide models or tools that allow assessing quantitatively the impact of information systems on patient pathway, the aim of this work is to propose a generic modelling framework for describing both patient pathway and the exchange of information between health institutions in the context of patient's care, concerning a given chronic pathology. Such framework should be compact and simple enough to be used by any stakeholder to build the specific model of a patient pathway including all HIS available in the environment. The resulting model can be used to support risk analysis and discrete event simulation, in order to assess the impact of health institutions' communication on the patient care pathway.

A Generic Modelling Framework for Describing the Management System of Information Exchanged between Health Institutions within the Context of Patient Pathway

In the perspective of assessing the impact of interactions between the hospital and the city health facilities on patient's cancer pathway, we have decided to implement a generic modelling framework for describing the management system of medical information which are exchanged between health institutions within the context of healthcare for patients suffering from a chronic disease. This modelling framework uses the process approach as a modelling method, as well as the BPMN (Business Process Model and Notation) as a graphical formalism with, however, some modifications made on that formalism. Furthermore, any model built starting from the proposed modelling framework will be represented according to two views : (i) a « Patient pathway » view and (ii) a « Shared health information system » view. The aim of such modelling structure is to increase the readability of the model and therefore the intelligibility of the described system.

(i) The « Patient pathway » view: described on one hierarchical level, it makes explicit all healthcare processes that can compose a patient pathway regarding a given disease (e.g. medical consultation, biological analysis examination, surgery, etc.) as well as the articulating of these processes, i.e. the various possibilities of process sequences. In order to illustrate our point, we will give the example of the patient's cancer pathway (see Fig. 1). The figure below describes the cancer patient's pathway through the processes that compose it: the red area constitutes the in-hospital care, while the green area will constitute the out-hospital care (within city health facilities).

Fig. 1. The cancer patient's pathway

(ii) The « Shared health information system » view: described for its part on three hierarchical levels:

Level 3: representing the highest level of abstraction, it allows describing for each process, implemented within a given health facility, its potential interactions (or information exchanges) with all processes implemented in other health facilities, knowing that the transmission of information can be done directly between two processes through a direct communication interface (e.g. postal mail, electronic messaging, fax, etc.), or indirectly by feeding an electronic patient record shared between health facilities within the health territory. Let's take as example the "hospital's medical consultation" process which interacts with two other processes occurring outside the hospital (see Fig. 2).

Fig. 2. Information exchanges between healthcare processes

We can see that each process has four inputs and four outputs. An input (I/P0) for the "Patient" entity, and three inputs for entities of "Information" type: I / P1 for the information received through a direct communication interface, I/P2 for the information that are available on an electronic patient record shared within the health territory, and I/P3 for the information that are already available on the patient record of the health facility which implements the described process. Idem for the outputs, there is one output (O/P0) for the entity "Patient" and three outputs for entities of "Information" type : O/P1 for information that has been exploited and which need to be (re)integrated into the patient record of the health facility, O/P2 for the information that will feed the electronic patient record shared within the health territory, and finally O/P3 for the information transmitted to another process via a direct communication interface (postal mail, fax...).

Level 2: describes each process from the view of the shared health information system, i.e. from the view of the management of medical information received or sent by the described process. The management of these information is performed through the concatenation of a number of dedicated activities which are (at most) in number of seven: (1) Receive an information, (2) Integrate an external information into the patient record, (3) Retrieve the patient record, (4) archive the patient record, ( 5) Consult an external information, ( 6) Produce an information and ( 7) Communicate an information (see Fig. 4). We can distinguish in that figure the four inputs and the four outputs of the process. We can also distinguish two types of transitions: the ones with an arrow in solid line correspond to patient flows, while dotted arrows represent information flows. Furthermore, we can see that the execution of some activities may require synchronization between the "patient" entity and an "information" entity such as for the activity "Consult an information".

Moreover, it is important to emphasise the generic nature of that level which allows describing any healthcare process from the point of view of the shared health information system, regardless the degree of development of the facility's information system (a paper or an electronic patient record, a non integrated or a highly integrated computer system, etc.).

Level 1: finally, this level allows describing the implementing procedures of each activity composing a given process: each activity has at least one implementing procedure, each procedure is characterized by a delay and a cost related to the mobilization of human, hardware, software and network resources which are necessary for its achievement. Example: The implementing procedures of the activity "Communicate an information" for the process "Medical consultation" within a hospital having an electronic patient record (see Fig. 3).

Conclusion

In this paper we proposed a new framework based on the process approach for patient pathway and information modelling. We first defined the context of the study in the e-SIS project. Then we presented the basics of the modelling framework which allows describing the management system of information exchanged between health institutions within the context of patient pathway. The resulting models will be used to support simulation for performance evaluation.

In future work we are currently working on the development of a rigorous methodology to propose a performance evaluation of the health information system on the regional level. Such quantitative evaluation will be performed taking into account (i) patient quality of care, (ii) working conditions of health-care professionals, and (iii) economic evaluation of health structures.

Fig. 3 .Fig. 4 .

 34 Fig. 3. The implementing procedures of the activity "Communicate an information" within the context of a medical consultation" at a hospital

Acknowledgments. The authors would like to thanks practitioners from the Centre Léon Bérard: L. Perrier, P. Biron, T. Durand; CH Sens: S. Aloui; Centre Jean Perrin: A. Doly.

This work is a part of a research project entitled e-SIS ("Assessment in Health Information Systemsfunded by the French