Serge Lallemand

Jean-Claude Sibuet

J.-M Auzende

J.-P Foucher

TECTONIC IMPLICATIONS OF CANYON DIRECTIONS OVER THE NORTHEAST ATLANTIC CONTINENTAL MARGIN 1

V JEAN CHARCOT since 1977.

MAIN PHYSIOGRAPHIC PROVINCES

ON CONTINENTAL MARGINS

The northeast Atlantic continental margins can be divided into six main domains (Figure 1) on the basis of the following criteria:

( Within sub zone 5b, the North Spanish margin is globally oriented E-W, and no major peak corresponds to the gravity directions. This area displays a broad variety of canyon trends (Figure 10).

Seven hundred and fifty measurements of orientation and length of segments of canyon axes have been

Fig. 4 .

 4 Fig. 3. Diagram of normalized number of events as a function of the direction of canyons, for area 1 defined in Figure 1.

Figure

  Figure 5 shows that the main peaks occur from-5' to 40' and from 45' to 65' for both sub zones 2b and 2c. These peaks are linked to the directions of late Hercynian faults in Brittany and on the continental shelf, directions which vary from NNE-SSW to E-W in azimuth. The major peak at N20' corresponds to the gravity trends but also to the direction of transecting faults located in the Meriadzek area by Montadert et al. [1979a] and reactivated during the

  Fig. 8. Diagram of normalized number of events as a function of the direction of canyons, for area 3 defined in Figure 1.

From

  Fig. 9. Diagram of normalized number of events as a function of the direction of canyons, for area 4 defined in Figure 1.

Figure

  Figure 11 shows a great variety of directions which reflects the influence of a complex tectonic pattern, largely similar north and south of 40'N latitude in subzones 6a and 6b, while the physiographic character of these two subzones

Figure

  Figure 12 shows such a correspondence between peaks belonging to the late Hercynian fracture pattern on both the Celtic and North Iberian Peninsula margins. Each dark peak represents identified late Hercynian trends (Figure 5 and 10). Peaks in subzones 2c and 2a, 2b correspond respectively to peaks in subzones 5a and 5b. The peak at 55' (with respect to area 2) is common between subzones 2c and 5a and three peaks at -85', 50' and 82' (with respect to area 2)

As most of the Sea Beam data have been ob- tained on the continental margin from Porcupine Seabight to the south of the

	Iberian	Peninsula,	a precise	picture	of
	the continental		slope is given.	A statis-
	tical	analysis	of the canyons,

based on 750 measurements, reveals that many of the canyons present sharp changes in their direction, indicating a structural control mainly linked to the late Hercynian trends, especially around the Iberian Peninsula. Nevertheless, the paths of canyons may merely reflect recent gravity processes, as in the Porcupine Seabight. Canyons locally follow the directions of listric and associated transecting faults (Permian to Triassic and upper Jurassic tools which provide precise information on morphologic trends. Twenty Sea Beam cruises or transits have been carried out, mainly on continental margins, which allow us to produce a new detailed bathymetric map (Plate 1) [Lallemand et al., 1985 a,b]. Most of the Sea Beam data have been acquired along the continental margins (Plate 2), which provides a fairly good picture of the morphology of the northeast Atlantic continental margins, especially along the slopes. Many of the canyons cross the margins obliquely and present sharp changes in their directions. This indicates that the canyon trends are not only controlled by gravity processes, but they are also largely influenced by the structural pattern [e.g., Boil lot et al., 1974; Ken.on et al., 1978; Sibuet and Berthois, 1979J. We propose to perform a statistical study of the direction of canyons and to correlate these directions with the structural pattern known on land and on the continental shelf. GEOLOGICAL EVOLUTION OF THE NORTH ATLANTIC OCEAN Tensional episodes occured from Permian to Lias in the areas of the future plate boundaries between Europe, North America, and Iberia. The continental crust was thinned, sometimes with formation of very limited amount of oceanic crust [e.g., Foucher et al., 1982], but the complete structural outline of all continental margins of the northeast Atlantic Ocean was mainly acquired during the late Jurassic-early Cretaceous phase of rifting which occured simultaneously with the opening of the central Atlantic Ocean. The oldest magnetic anomalies identified in the northeast Atlantic Ocean are MO (late Aptian) west of Galicia Bank and 34 (upper Santonian) in the Bay of Biscay and west of the British Isles [Sibuet and Ryan, 1979; Guennoc et al., 1978, 1979] (Figure 1). Nevertheless,

	oceanic	crust	still	exists		between	the
	oldest identified	magnetic anomalies and
	the continent-ocean	boundary. From
	drilling	results	the first		appearance of
	oceanic	crust	has been dated	lower	Albian
	at Goban Spur •Masson et al., 1984], early
	Albian	near	the Meriadzek	terrace	in the

northern Bay of Biscay [Montadert et al., 1979a,b; Pastouret et al., 19•81]

and late Aptian west of Galicia Bank [Sibuet and Ryan, 1979]. This indicates that the initial emplacement of oceanic crust propagated from south to north. The formation of the Bay of Biscay results from the rotation of Iberia with respect to Europe around a pole located in southwest Europe •L•e Pichon et al., 1971; Olivet et al., 1984J from late Jurassic-early Cretaceous to the time of anomaly 33 (Campanian) ESibuet and Ryan, 1979] or during the Cretaceous Quiet Magnetic Zone (Albian to Santonian) for Olivet et al. [1984]. No conclusive evidence from land geology in the Pyrenees and Aquitaine Basin allows us to discriminate between these two hypotheses. Simultaneously, the North Atlantic opening was going on. From late Cretaceous to late Eocene, the compressive movements resulted in the uplift of Pyrenees and the format ion of the North Spanish subduct ion zone with a paroxysmal phase during Eocene [Sibuet and Le Pichon, 1971; Le Pichon and Sibuet, 1971a].

margin, oriented approximately NllO ø, resulted from the southeastward movement of Iberia with respect to Eurasia in lower Cretaceous [e.g., Le Pichon et al., 1971]. Area 3. The Armorican continental margin, oriented N135', was also formed during the rotation of Iberia but corresponds to a sheared continental margin [Le Pichon et al., 1971]. It presents a narrower continental slope than the Celtic margin. Area 4. The Landes Plateau is bounded to the north and south by the Cap Ferret Canyon and the Gouf of Cap Breton, respectively, which are both oriented 50 ø 45 ø 15 ø 10 o 5 ø \ / •D Area 5 51) / • / 20 ø 15 o 10 o 5 ø Fig. 1. Simplified structural map of the northeast Atlantic region. The main physiographic provinces of the continental margin and the location of canyons and their tributaries used in this study are shown. Identified magnetic anomalies from Guennoc et al. [1978, 1979]. 1, North Iberia Paleogene thrusting front; 2, Late Hercynian faults [Parga, 1969; Arthaud and Matte, 1975; Pegrum and Mounteney, 1978; Montadert et al., 1979a; Sibuet et al., 1948b]. 50 ø 45 ø ÷40 ø N1OO ø, and to the west by the Santander Canyon which is roughly north-south in trend. Area 5. The North Spanish continental margin is the counterpart of the Celtic margin but was largely tectonized during the lower Tertiary compressire phase (100-150 km of amplitude) which gave rise to the formation of the North Spanish marginal 9ø00 ' 48,oL oo[-CELTIC M•RGI• SHAMROCK CANYON 20km I .••• Generol ,rendrgOlf n ?.000 • •, ...... 8o40 ' ' I 1000 47 ø 40' 9000 ' 8o40 ' Fig. 2. Straight lines corresponding to linear segments of the Shamrock canyon axis and its tributaries superimposed on the Sea Beam map EPastouret et al., 1982] located as box A on Figure 1. Depths are in uncorrected meters (15OO m/s). Isobath spacing 20 m. 47 ø 40' 8020 ' trough [Sibuet and Le Pichon, 1971; Sibuet et al., 1971]. Area 6. The west Iberian continental margin which is roughly oriented north-south presents, in its northern portion, a large promontory named Galicia Bank (subarea 6a) that is separated from the continental shelf by the Interior Basin filled up with sediments [Auzende et al., 1979]. METHODOLOGY Most of the tectonic lines which appeared during the formation of the continental margins represent preferential erosional guides for incipient channels and canyons. Nevertheless, especially during the Plio-Quaternary period, important sedimentary gravity processes occured along the major gradient of slopes, giving rise to canyons oriented perpendicularly to the margin. A statistical analysis of the orientation of the canyons was undertaken in order to discriminate between tectonic and gravity directions and to compare these tectonic trends with onland and offshore structural trends.

	Area	2.	The Celtic	continental

the formation of the Bay of Biscay is the one initially proposed by Le Pichon et al. [1970, 1971]. Since that time, several authors have proposed slight modifications which concern the timing, the pole position or the rotation angle [e.g., Le Pichon and Sibuet, 1971b; Choukroune et al., 1973; Olivet et al., 1984; Savostin et al., 1986]. Published angles of rotations vary from 30' [Le Pichon et al., 1971]•to 23' [Le Pichon and Sibuet, 1971bJ. Olivet et al. [1984] adopted an angle of 24'. Values of this angle depend on the position of the rotation pole but also on the choice of fitted isobaths or features. In all these reconstructions, the North Spanish margin was initially connected to the Celtic margin. It is tempting to look at the angular correlation between lineaments predating the opening of the Bay of Biscay: in this case, the late Hercynian fracturation pattern even if it was reactivated later. We saw indeed that the geological and tectonic evolutions of both margins differ considerably after their creation and separation, especially during the Eocene compressire phase which affected the southern margin. Consequent ly, correlation between peaks must not be done on a criterion of amplitude, but on a criterion of global correspondence of peaks without the influence of gravity processes.

	ARGUMENTS IN FAVOR OF A 23' ROTATION
	OF IBERIA	WITH	RESPECT	TO EUROPE
	The currently	accepted hypothesis about

ø

12, 83-88, 1971a.Le Pichon, X., and J.-C. Sibuet, Comments

Montadert,

The


Roland Bacher

A new inequality for the Hermite constants

Keywords: Lattice packing, Hermite constant, inequality

We describe continuous increasing functions C n (x) such that γ n ≥ C n (γ n-1 ) where γ m is Hermite's constant in dimension m. This inequality yields a new proof of the Minkowski-Hlawka bound ∆ n ≥ ζ(n)2 1-n for the maximal density ∆ n of n-dimensional lattice-packings. 1

Introduction and main results

We denote by min(Λ) = min λ∈Λ\{0} λ, λ the minimum (defined as the squared Euclidean length of a shortest non-zero element) of an n-dimensional lattice Λ ⊂ E n in the Euclidean vector-space E n and define the density of Λ by

∆(Λ) = (min Λ) n 4 n det Λ V n
where V n = π n/2 (n/2)! stands through the whole paper for the volume of the n-dimensional unit-ball in E n . The density ∆(Λ) is the proportion of volume occupied by a maximal open Euclidean ball embedded in the flat torus E n /Λ with volume det(Λ) and having a shortest closed geodesic of length min(Λ). The largest density ∆ n = ∆(Λ n ) achieved by an n-dimensional lattice Λ n is called the maximal density in dimension n. Related constants are the maximal center density δ n = ∆ n /V n and the Hermite constant γ n = 4δ 2/n n in dimension n. The sequence γ 1 , γ 2 , . . . of Hermite constants satisfies for n ≥ 3 Mordell's inequality

γ n ≤ γ (n-1)/(n-2) n-1
which yields an upper bound for γ n (if n ≥ 3) in terms of γ n-1 . Our main result is a complementary inequality bounding γ n from below in terms of γ n-1 . For the convenience of the reader we state it in three equivalent ways, either in terms of densities ∆ m , center-densities δ m or Hermite constants γ m in dimension m. It involves the Möbius function µ : N >0 -→ Z defined by µ(l) = (-1) a for a natural integer l ∈ N which is a product of a distinct primes and by µ(l) = 0 if l is divisible by the square of a prime number. 1 Theorem 1.1 (i) The maximal densities ∆ n-1 and ∆ n of lattice-packings in dimensions n -1 and n ≥ 2 satisfy the inequality

2 n-1 ∆ n-1 ⌊2∆nV n-1 /(∆ n-1 Vn)⌋ k=1   l|k µ(l) l n-1   1 - k∆ n-1 V n 2∆ n V n-1 2 (n-1)/2 ≥ 1
where the sum l|k is over all positive integral divisors l ∈ N of the natural integer k.

(ii) The maximal center densities δ n-1 and δ n of lattice-packings in dimensions n -1 and n ≥ 2 satisfy the inequality

2 n-1 δ n-1 π (n-1)/2 ((n -1)/2)! ⌊2δn/δ n-1 ⌋ k=1   l|k µ(l) l n-1   1 - kδ n-1 2δ n 2 (n-1)/2 ≥ 1.
(iii) The Hermite constants γ n-1 and γ n in dimensions n-1 and n ≥ 2 satisfy the inequality

π (n-1)/2 ((n -1)/2)! ⌊γ n/2 n /γ (n-1)/2 n-1 ⌋ k=1   l|k µ(l) l n-1   γ n-1 -k 2 γ n-1 γ n n (n-1)/2 ≥ 1.
Consider the function

F n (x, y) = ⌊ √ x y⌋ k=1 l|k µ(l) l n-1 x - k y 2 (n-1)/2
, defined for x, y > 0. It is increasing in y, strictly increasing in y for y > 1 √ x , continuous and extends continuously to y = 0 since F n (x, y) = 0 for y ≤ 1 √ x . There exists thus a continous function x -→ Y n (x) such that

F n (x, Y n (x)) = ((n -1)/2)! π (n-1)/2
for all x > 0. Set

C n (x) = sup 0<ξ≤x (ξ (Y n (ξ)) 2/n ).
Theorem 1.2 Let γn-1 ≤ γ n-1 be a lower bound for Hermite's constant in dimension n -1. Then γ n ≥ C n (γ n-1 ) .

Analogous results hold of course for ∆ n = 2 -n γ n/2 n V n and δ n = 2 -n γ n/2 related to Hermite's constant γ n = 4(∆ n /V n ) 2/n = 4δ 2/n n . Remark 1.3 (i) The inequality of Theorem 1.2 is tight for n = 2. For n = 3, we get from δ 2 = 1/2 √ 3 the lower bound 0.1695 ≤ δ 3 = 1/4 √ 2 ∼ 0.1768. For n = 9, the known value δ 8 = 1/16 gives the lower bound δ 9 ≥ 0.0388 (a lattice with center-density 0.0442 is known), for n = 25 the known value δ 24 = 1 coming from the Leech lattice (see Cohn-Kumar, [START_REF] Cohn | The densest lattice in twenty-four dimensions[END_REF] and [START_REF] Cohn | Optimality and uniqueness of the Leech lattice among lattices[END_REF]) yields δ 25 ≥ 0.657 (a lattice with center-density 0.707 is known).

(ii) The above examples show that our inequality is better than the trivial inequality

δ n ≥ µ n n-1 4 n det(Λ n-1 ⊕ µ n-1 Z) = δ n-1 2
obtained by considering the orthogonal sum

Λ n-1 ⊕ µ n-1 Z of a densest (n - 1)-dimensional lattice Λ n-1 with minimum µ n-1 = min λ∈Λ n-1 \{0} λ, λ . (iii) The factor l|k µ(l) l n-1 = p prime,p|k 1 - 1 p n-1
yields only a minor improvement for huge n and is the analogue of a standard trick leading to the factor

ζ(n) in the Minkowski-Hlawka bound ∆ n ≥ ζ(n) 2 n-1 . (iv) The application x -→ C n (x) of Theorem 1.
2 is strictly increasing for x huge enough. The computations of Section 6 show in fact that one has C n (x) = x(Y n (x)) 2/n (at least for huge n) for all values of x which are of interest.

Remark 1.4 Starting with the inequality of assertion (ii) in Theorem 1.1, and using l|k µ(l)

l n-1 = p|k 1 -1 p n-1
< 1 (where the product is over all prime divisors of k), A. Marin pointed out the easy inequalities

1 ≤ 2 n-1 δ n-1 π (n-1)/2 ((n -1)/2)! ⌊2δn/δ n-1 ⌋ k=1   l|k µ(l) l n-1   1 - kδ n-1 2δ n 2 (n-1)/2 ≤ 2 n δ n V n-1 ⌊2δn/δ n-1 ⌋ k=1 1 - kδ n-1 2δ n 2 (n-1)/2 δ n-1 2δ n ≤ 2 n δ n V n-1 1 0 1 -x 2 (n-1)/2 dx = 2 n δ n V n 2 = 2 n-1 ∆ n which show ∆ n ≥ 1 2 n-1
. This is, up to the factor ζ(n), the Minkowski-Hlawka bound for ∆ n .

The following, technically more involved result yields a slightly better lower bound: Theorem 1.5 For all ǫ > 0, there exists N such that

∆ n ≥ 1 -ǫ 2 n ∞ k=1 e -k 2 π ∼ (1 -ǫ) 23.1388 2 -n for all n ≥ N .
Remark 1.6 Theorem 1.5 is slightly better that the Minkowski-Hlawka bound which shows the existence of lattices with density at least ζ(n) 2 1-n , cf. formula (14) in [START_REF] Conway | Sphere packings, Lattices and Groups[END_REF], Chapter 1. The best known bound for densities achieved by lattice packings (together with a very nice proof ) seems to be due to Keith Ball and asserts the existence of n-dimensional lattices with density at least 2(n -1)2 -n ζ(n), see [START_REF] Ball | A lower bound for the optimal density of lattice packings[END_REF]. Previously, similar results where proven by Rogers and Davenport-Rogers, see [START_REF] Rogers | Existence theorems in the geometry of numbers[END_REF] and [START_REF] Davenport | Hlawka's theorem in the geometry of numbers[END_REF]. Somewhat related are also [START_REF] Krivelevich | A lower bound on the density of sphere packings via graph theory[END_REF] and [START_REF] Gaborit | On the construction of dense lattices with a given automorphism group[END_REF] which describe elementary constructions of dense sphere and lattice packings.

The paper is organized as follows: Section 2 introduces notations and summarizes for the convenience of the reader a few well-known facts on lattices. It contains also an easy (and seemingly not very well-known) result on integral sublattices which are orthogonal to a non-zero integral vector in Z n+1 .

In Section 3 we define of µ-sequences which are the main tool of this paper. Theorem 3.3 of this Section gives a quantitative (and somewhat technical) statement for extending a suitable finite µ-sequence (s 0 , . . . , s n-1 ) to a µ-sequence (s 0 , . . . , s n-1 , s n ). The associated n-dimensional lattice (s 0 , . . . , s n ) ⊥ ∩ Z n+1 is obtained by a close analogue of lamination with respect to its (n -1)-dimensional sublattice (s 0 , . . . , s n-1 ) ⊥ ∩ Z n . Theorem 3.3 is the central result of this paper since it implies easily Theorem 1.1 and Theorem 1.2 as shown at the end of Section 3. The proof of Theorem 1.5 is more technical and given in Section 6.

Section 4 states and proves a weaker and easier statement than Theorem 3.3. Although not necessary for the other parts of the paper, this section describes a fairly elementary and almost effective method for constructing dense lattices. It contains moreover the essence of the main idea for proving Theorem 3.3.

Section 5 describes the proof of Theorem 3.3. Section 6 is devoted to the proof of Theorem 1.5. Section 7 contains a few final comments and remarks.

Definitions

All facts concerning lattices needed in the sequel are collected in this Section for the convenience of the reader, see [START_REF] Conway | Sphere packings, Lattices and Groups[END_REF] and [START_REF] Martinet | Perfect Lattices in Euclidean Spaces[END_REF] for more on lattices and lattice-packings.

An n-dimensional lattice is a discret-cocompact subgroup Λ of the n-dimensional Euclidean vector space E n . Denoting by , the scalar product and choosing a Z-basis b 1 , . . . , b n of a lattice Λ = ⊕ n j=1 Zb j , the positive definite symmetric matrix G ∈ R n×n with coefficients

G i,j = b i , b j
is a Gram matrix of Λ. Its determinant det(G), called the determinant of Λ, is independent of the choosen basis b 1 , . . . b n and equals the squared volume of the flat torus E n /Λ. The norm of a lattice vector λ ∈ Λ is defined as λ, λ and equals thus the square of the Euclidean norm λ, λ . A lattice Λ is integral if all scalar products { λ, µ | λ, µ ∈ Λ} are integral. An integral lattice of determinant 1 is unimodular. An Euclidean lattice Λ is unimodular if and only if every group homomorphism ϕ : Λ -→ Z is of the form ϕ(v) = v, w ϕ for a suitable fixed element w ϕ ∈ Λ. The minimum min Λ = min λ∈Λ\{0} λ, λ of a lattice Λ is the norm of a shortest non-zero vector in Λ. The density ∆(Λ) and the center-density δ(Λ) of an n-dimensional lattice Λ are defined as

∆(Λ) = (min Λ) n 4 n det Λ V n and δ(Λ) = (min Λ) n 4 n det Λ
where V n = π n/2 /(n/2)! denotes the volume of the n-dimensional unit-ball in E n . These two densities are proportional for a given fixed dimension n and ∆(Λ) measures the (asymptotic) proportion of space occupied by the sphere packing of Λ obtained by centering n-dimensional Euclidean balls of radius min Λ/4 at all points of Λ. Given an n-dimensional lattice Λ ⊂ E n the subset

Λ ♯ = {x ∈ E n | x, λ ∈ Z ∀λ ∈ Λ}
is also a lattice called the dual lattice of Λ. The scalar product induces a natural bijection between Λ ♯ and the set of homomorphisms Λ -→ Z. A lattice Λ is integral if and only if Λ ⊂ Λ ♯ . For an integral lattice, the determinant group Λ ♯ /Λ is a finite abelian group consisting of (det Λ) elements.

A sublattice M ⊂ Λ is saturated if Λ/M is without torsion or equivalently if M = (M ⊗ Z R) ∩ Λ. A sublattice Λ ⊂ M is thus saturated if and only if M is a direct factor of the additive group M .
The following result is well-known (cf. Chapter I, Proposition 9.8 in [START_REF] Martinet | Perfect Lattices in Euclidean Spaces[END_REF]):

Proposition 2.1 Let M and N be two saturated sublattices of dimension m and n in a common unimodular lattice Λ of dimension m + n such that M and N are contained in orthogonal subspaces.

Then the two determinant groups M ♯ /M and N ♯ /N are isomorphic. In particular, the lattices M and N have equal determinants.

Proof Since Λ = Λ ♯ is unimodular, orthogonal projection Λ -→ M ♯ yields a surjective homomorphism from Λ onto M ♯ with kernel (N ⊗ Z R) ∩ Λ coinciding with N since N is saturated. This shows M ♯ ∼ Λ/N and thus M ♯ /M ∼ Λ/(M ⊕ N ). Exchanging the role of M and N implies the result.

2

Two lattices Λ and M are similar, if there exists a bijection Λ -→ M which extends to an Euclidean similarity from Λ ⊗ Z R to M ⊗ Z R. The set of similarity classes of lattices is endowed with a natural topology: a neighbourhood of an n-dimensional lattice Λ is given by all lattices having a Gram matrix in

R >0 V (G) where V (G) ⊂ R n×n is a neighbourhood of a fixed Gram matrix G of Λ.
Similar lattices have identical densities and the density function Λ -→ ∆(Λ) is continuous with respect to the natural topology on similarity classes.

Consider the set

L n = {Λ(s) | s ∈ N n+1 \ {0}} of all n-dimensional integral lattices of the form Λ(s) = {z ∈ Z n+1 | z, s = 0}
where s ∈ N n+1 \ {0} is a non-zero vector of length n + 1 with non-negative integral coordinates. where L = (l i,j ) 1≤i,j≤n is an invertible lower triangular matrix. Choose a large real number κ > 0 and consider the integral lower triangular matrix L(κ) whose coefficients li,j ∈ Z satisfy

| li,j -κl i,j | ≤ 1/2
and are obtained by rounding off each coefficient of κL to a nearest integer.

Define the integral matrix

B(κ) =       l1,1 1 0 0 . . . l2,1 l2,2 1 0 . . . . . . . . . ln,1 ln,2 . . . ln,n 1       of size n × (n + 1) with coefficients b i,j =      li,j if j ≤ i 1 if j = i + 1 0 otherwise .
The rows of B(κ) span an integral sublattice Λ(κ) of dimension n in Z n+1 . Moreover, the lattice Λ(κ) is saturated since deleting the first column of B(κ) yields an integral unimodular square matrix of size n × n. The special form of B(κ) shows that there exists an integral column-vector

v(κ) =       1 -l1,1 l1,1 l2,2 -l2,1 . . .       ∈ Z n+1 such that B(κ)v(κ) = 0. We have thus Λ(κ) = v(κ) ⊥ ∩ Z n+1 ⊂ E n+1 .
Since lim κ→∞ 1 κ B(κ) is given by the matrix L with an extra column of zeros appended, we have

lim κ→∞ 1 κ 2 B(κ)(B(κ)) t = G
and the lattice 1 κ Λ(κ) converges thus to the lattice Λ for κ → ∞. Considering the integral vector s = (s 0 , s 1 , . . .) ∈ N n+1 defined by s i = |v(κ) i+1 | for i = 0, . . . , n, we get an integral lattice

{z = (z 0 , . . . , z n ) ∈ Z n+1 | z, s = 0} of L n which is isometric to Λ(κ). 2 3 µ-sequences
Let µ ≥ 2 be a strictly positive integer. A µ-sequence is a finite or infinite sequence s 0 = 1, s 1 , s 2 , . . . of (l + 1) strictly positive integers such that the n-dimensional lattice

Λ n = {(z 0 , z 1 , . . . , z n ) ∈ Z n+1 | n k=0 s k z k = 0} = (s 0 , . . . , s n ) ⊥ ∩ Z n+1
has minimum ≥ µ for all n ≥ 1 which make sense (ie. for n ≤ l if the sequence (s 0 , s 1 , . . . , s l ) has finite length l). The letter µ in a µ-sequence stands for minimum and should not be confused with the Moebius-function, (unfortunately also) denoted µ : N -→ {-1, 0, 1}. Since Λ n is saturated in Z n+1 by construction and orthogonal to the 1-dimensional saturated lattice Z(s 0 , . . . , s n ) ⊂ Z n+1 , Proposition 2.1 shows that we have det(Λ n ) = n k=0 s 2 k . We get thus a lower bound for the density

∆(Λ n ) = (min Λ n ) n 4 n det Λ n V n ≥ µ n 4 n n k=0 s 2 k V n
of the n-dimensional lattice Λ n associated to a µ-sequence (s 0 , . . . , s n , . . .). This lower bound is an equality except if the sequence (s 0 , . . . , s n ) is a (µ + 1)-sequence.

Remark 3.1 We hope that the double meaning of µ will not confuse the reader: µ(l) ∈ {-1, 0, 1} denotes always the Möbius function of a natural integer l while µ or µ 1 , µ 2 , . . . stands for natural integers. Remark 3.2 (i) The condition s 0 = 1 ensures that (s 0 , . . . , s n ) ⊂ Z n+1 generates a saturated 1-dimensional sublattice and will be useful for proving Lemma 5.1. It can however be weakened by requiring that s 0 , . . . , s n are without common non-trivial divisor. Lemma 5.1 (which applies to a sequence of µ-sequences) remains valid if s 0 is uniformly bounded.

It is of course also possible (but not very useful) to consider sequences with coefficients in Z.

(ii) Any subsequence s i 0 = s 0 , s i 1 , s i 2 , . . . of a µ-sequence is again a µ-sequence and permuting the terms of a µ-sequence by a permutation fixing s 0 yields of course again a µ-sequence.

(iii) Lattices associated to µ-sequences are generally neither perfect nor eutactic (cf. [START_REF] Martinet | Perfect Lattices in Euclidean Spaces[END_REF] for definitions). Their densities can thus generally be improved by suitable deformations. Theorem 3.3 Let µ 1 , µ 2 , . . . be a strictly increasing sequence of natural integers 2 ≤ µ 1 < µ 2 < . . .. Suppose that we have finite µ r -sequences (s(µ r ) 0 , . . . , s(µ r ) n-1 ) with existing limit-density

∆n-1 = lim r→∞ µ (n-1)/2 r 4 n-1 n-1 i=0 s(µ r ) 2 i V n-1 > 0
for the sequence of lattices (s(µ r ) 0 , . . . , s(µ r ) n-1 ) ⊥ ⊂ Z n . Let σ n be a positive real number such that

2 n-1 ∆n-1 A k=1   l|k µ(l) l n-1   1 -k 2 2 n-1 ∆n-1 V n V n-1 σ n 2 (n-1)/2 < 1
where

A = 2 1-n V n-1 V n ∆n-1 σ n .
Then there exists a natural integer R such that for all r ≥ R, the µ r -sequence (s(µ r ) 0 , . . . , s(µ r ) n-1 ) can be extended to a µ r -sequence (s(µ r ) 0 , . . . , s(µ

r ) n ) satisfying 0 < s(µ r ) n < σ n µ n/2 r V n .
The proof of Theorem 3.3 will be given in section 5. Assuming Theorem 3.3, we proceed now to prove Theorem 1.1.

Proof of Theorem 1.1 and 1.2

Suppose that the inequality of assertion (i) does not hold for some natural integer n. By Proposition 2.2 we can find a sequence of finite µ r -sequences (s(µ r ) 0 , . . . , s(µ r ) n-1 ) (with µ r → ∞) such that lim r→∞ µ

(n-1)/2 r 4 n-1 n-1 i=0 s(µ r ) 2 i V n-1 = ∆ n-1 .
Consider the function

x -→ 2 n-1 ∆ n-1 A(x) k=1   l|k µ(l) l n-1   1 -k 2 2 n-1 ∆ n-1 V n V n-1 x 2 (n-1)/2
where

A(x) = ⌊ 2 1-n V n-1
Vn∆ n-1 x ⌋ for x > 0. We claim that this function is continuous and strictly decreasing in x. Indeed, for increasing x ∈ [0,

2 1-n V n-1 kVn∆ n-1 ], a summand   l|k µ(l) l n-1   1 -k 2 2 n-1 ∆ n-1 V n V n-1 x 2 (n-1)/2
decreases continuously from l|k µ(l)

l n-1 = p|k 1 -1 p n-1 > 0 to 0. Such a summand disappears if it becomes zero and their number A(x) = ⌊ 2 1-n V n-1
Vn∆ n-1 x ⌋ is finite and decreases with x.

This shows that we can choose a positive real number σ n < 1 2 n ∆n such that we have the inequalities

2 n-1 ∆ n-1 ⌊2∆nV n-1 /(∆ n-1 Vn)⌋ k=1   l|k µ(l) l n-1   1 - k∆ n-1 V n 2∆ n V n-1 2 (n-1)/2 < < 2 n-1 ∆ n-1 A k=1   l|k µ(l) l n-1   1 -k 2 2 n-1 ∆ n-1 V n V n-1 σ n where A = ⌊ 2 1-n V n-1
Vn∆ n-1 σn ⌋. Applying Theorem 3.3 and extracting a suitable subsequence from µ 1 , µ 2 , . . ., we can suppose that all sequences (s(µ r ) 0 , . . . , s(µ r ) n-1 ) can be extended to µ r -sequences (s(µ r ) 0 , . . . , s(µ r ) n ) with

lim r→∞ s(µ r ) n µ n/2 r = α ≤ σ n V n . Using lim r→∞ 1 µ n-1 r n-1 i=0 s(µ r ) 2 i = V 2 n-1 4 n-1 ∆ 2 n-1
we have

lim r→∞ 1 µ n r n i=0 s(µ r ) 2 i = lim r→∞ 1 µ r V 2 n-1 4 n-1 ∆ 2 n-1 + α 2 = α 2
and get the existence of a sequence of n-dimensional lattices

(s(µ r ) 0 , . . . , s(µ r ) n ) ⊥ ⊂ Z n+1
with limit-density

lim r→∞ µ n r 4 n n i=0 s(µ r ) 2 i V n = 1 2 n α V n ≥ 1 2 n σ n V n V n > ∆ n in contradiction with maximality of ∆ n . 2 Proof of Theorem 1.2 Choose ξ < x such that C n (x) = ξ Y n (ξ) 2/n .
The inequality ξ ≤ γ n-1 implies the existence of an (n -1)-dimensional lattice with density 2 -(n-1) ξ (n-1)/2 V n-1 . Theorem 3.3 implies the existence of n-dimensional lattice with density 2 -n (C n (x)) n/2 V n . 2

4 An easy crude bound for the lexicographically first µ-sequence Theorem 4.1 Given an integer µ ≥ 2 there exists an increasing µ-sequence s 0 = 1 ≤ s 1 ≤ . . . such that

s n ≤ 1 + µ -2 µ -1 + n/4 n √ π n (n/2)! ≤ √ µ µ + n/4 n √ π n (n/2)! for all n ≥ 1.
The proof of Theorem 4.1 is elementary and consists of an analysis of the "greedy algorithm" which constructs the first µ-sequence with respect to the lexicographic order on sequences. An easy analysis shows that the lexicographically first µ-sequence satisfies the first inequalities of Theorem 4.1. The greedy algorithm, although very simple, is unfortunately useless for practical purposes.

A µ-sequence satisfying the inequalities of Theorem 4.1 yields already rather dense lattices as shown by the next result.

Corollary 4.2 For any µ ≥ 2, there exists a µ-sequence (s 0 , s 1 , . . . , s n ) ∈ Z n+1 such that the density of the associated lattice

Λ n = (s 0 , . . . , s n ) ⊥ ∩Z n+1 satisfies ∆(Λ n ) ≥ (1 + n/(4µ)) -n/2
2 n (n + 1)µ .

Remark 4.3 Taking µ ∼ n 2 /4 we get for large n the existence of n-dimensional lattices with density ∆ roughly at least equal to

1 2 n-1 n (n + 1) e which is reasonably close to the Minkowski-Hlawka bound ∆ n ≥ ζ(n) 2 1-n .
Lemma 4. [START_REF] Cohn | Optimality and uniqueness of the Leech lattice among lattices[END_REF] The standard Euclidean lattice Z n contains at most

2 µ + n/4 n π n/2 (n/2)!
vectors of (squared Euclidean) norm ≤ µ.

Proof We denote by

B ≤ √ ρ (x) = {z ∈ E n | z -x, z -x ≤ ρ} the closed Euclidean ball with radius √ ρ ≥ 0 and center x ∈ E n . Given √ µ, √ ρ ≥ 0 and x ∈ B ≤ √ µ (0), the closed half-ball {z ∈ E n | z, x ≤ x, x } ∩ B ≤ √ ρ (x)
(obtained by intersecting the closed affine halfspace

H x = {z ∈ E n | z, x ≤ x, x } with the Euclidean ball B ≤ √ ρ (x) centered at x ∈ ∂H x ) is contained in B ≤ √ µ+ρ (0). Since the regular standard cube C = [- 1 2 , 1 2 ] n ⊂ E n
of volume 1 is contained in a ball of radius n/4 centered at the origin, the intersection

(z + C) ∩ {x ∈ E n | x, x ≤ µ + n/4} = (z + C) ∩ B ≤ √ µ+n/4 (0)
is of volume at least 1/2 for any element z ∈ E n of norm z, z ≤ µ.

Since integral translates of C tile E n , we have

1 2 ♯{z ∈ Z n | z, z ≤ µ} ≤ Vol {x ∈ E n | x, x ≤ µ + n/4} .
Using the fact that the unit ball in Euclidean n-space has volume π n/2 /(n/2)! (cf. Chapter 1, formula 17 in [START_REF] Conway | Sphere packings, Lattices and Groups[END_REF]) we get the result. 2 Proof of Theorem 4.1 For n = 0, the first inequality boils down to s 0 = 1 ≤ 1 + √ µ -2 and holds for µ ≥ 2. Consider now for n ≥ 1 a µ-sequence (s 0 , . . . , s n-1 ) ∈ N n .

Introduce the set

F n = {(a, k) ∈ N 2 | ∃ z = (z 0 , . . . , z n-1 ) ∈ Z n \ {0} such that
ak = z, (s 0 , . . . , s n-1 ) and z, z + k 2 < µ} .

Since Λ n-1 has minimum ≥ µ, the equality z, (s 0 , . . . , s n-1 ) = 0 implies z, z ≥ µ for z ∈ Z n \ {0}. This shows that we have a, k > 0 for (a, k) ∈ F n .

Since for a given pair of opposite non-zero vectors ±z ∈ Z n with norm 0 < z, z < µ there are at most µ -1z, z ≤ √ µ -2 strictly positive integers k such that z, z + k 2 < µ, such a pair ±z of vectors contributes at most √ µ -2 distinct elements to F n . The cardinality

f n = ♯(F n ) of F n is thus bounded by f n ≤ µ -2 ♯{z ∈ Z n | 0 < z, z ≤ µ -1} 2 ≤ µ -2 µ -1 + n/4 n π n/2 (n/2)!
where the last inequality follows from Lemma 4.4. There exists thus a strictly positive integer

s n ≤ f n + 1 ≤ 1 + µ -2 µ -1 + n/4 n π n/2 (n/2)! such that (s n , k) ∈ F n for all k ∈ N.
The strictly positive integer s n satisfies the first inequality of Theorem 4.1 and it is straightforward to check that the n-dimensional lattice

Λ n = {z ∈ Z n+1 | n i=0 s i z i = 0}
has minimum ≥ µ. This shows the first inequality. Choosing for s n the smallest strictly positive integer such that (s n , k) ∈ F n for all k ∈ N and iterating this construction yields clearly an increasing µ-sequence.

The second inequality ] n of volume 1. For µ ≥ 3 we have to establish the inequality Φ(1) -Φ(0) ≥ 1 where

1 + µ -2 µ -1 + n/4 n √ π n (n/2)! ≤ √ µ µ + n/4 n √ π n (n/
Φ(t) = µ -2 + 2t µ -1 + t + n/4 n √ π n (n/2)! .
We get thus

Φ(1) -Φ(0) ≥ inf ξ∈(0,1) Φ ′ (ξ) ≥ 1 √ µ µ -1 + n/4 n √ π n (n/2)! + n 2 √ µ -2 µ -1 + n/4 n-2 √ π n (n/2)! .
For n = 1 and µ ≥ 2 we have

Φ(1) -Φ(0) ≥ 1 - 1 µ √ π √ π/2 ≥ 2 √ 2 > 1 .
For n ≥ 2 and µ ≥ 3 we get

Φ(1) -Φ(0) ≥ 2 + n/4 n-2 √ π n-2 ((n -2)/2)! π
and the right-hand side equals π > 1 for n = 2. For n > 2, the right hand side equals π times the volume of the (n -2)-dimensional ball of radius 2 + n/4 containing the regular cube [-1 2 , 1 2 ] n-2 of volume 1. The second inequality follows.

2 Proof of Corollary 4.2 Theorem 4.1 shows the existence of a µ-sequence (s 0 = 1, . . . , s n ) satisfying

s 0 , . . . , s n ≤ √ µ µ + n/4 n √ π n (n/2)! .
This shows for the lattice Λ n = (s 0 , . . . , s n ) ⊥ ∩ Z n+1 the inequality

det Λ n = n i=0 s 2 i ≤ (n + 1)µ(µ + n/4) n π n ((n/2)!) 2 = (n + 1)µ(µ + n/4) n V 2 n and implies ∆(Λ n ) ≥ µ n 4 n (n + 1)µ(µ + n/4) n V 2 n V n which proves Corollary 4.2. 2
5 Proof of Theorem 3.3

The main idea for proving Theorem 3.3 is to get rid of a factor √ µ when computing an upper bound f for the size of the finite set F considered in the proof of Theorem 4.1. This is possible since the volume of the standard unit-ball of large dimension concentrates along linear hyperplanes. During the proof, we use for simplicity the slightly abusive notation µ = µ k and (s 0 , . . . , s n ) = (s(µ k ) 0 , . . . , s(µ k ) n ). Since µ belongs to the strictly increasing integral sequence µ 1 < µ 2 < . . . tending to infinity, we consider sequences in the µ → ∞ limit. This allows us to neglect boundary effects when replacing counting arguments by volume-computations.

In the sequel we write

g(x) ∼ x→α h(x) , respectively g(x) ≤ x→α h(x) , for lim x→α g(x) h(x) = 1 , respectively limsup x→α g(x) h(x) ≤ 1 ,
where g(x), h(x) > 0.

Proof of Theorem 3.3 We prove first a weaker statement assuming the stronger inequality

2 n-1 ∆n-1 A k=1 1 -k 2 2 n-1 ∆n-1 V n V n-1 σ n 2 (n-1)/2 < 1
where

A = ⌊ 2 1-n V n-1 V n ∆n-1 σ n ⌋.
Details for dealing with the extra factor l|k µ(l) l n-1 will be given later. Choose a positive real number σn < σ n such that we have the equalities

A = ⌊ 2 1-n V n-1 V n ∆n-1 σ n ⌋ = ⌊ 2 1-n V n-1 V n ∆n-1 σn ⌋
(where ⌊x⌋ ∈ Z denotes the integer part of x ∈ R) and the inequality

2 n-1 ∆n-1 A k=1 1 -k 2 2 n-1 ∆n-1 V n V n-1 σn 2 (n-1)/2 < 1.
We fix σn in the sequel and introduce ǫ = σn σn -1 > 0. We prove Theorem 3.3 for all µ huge enough by showing the existence of a µ-sequence (s 0 , . . . , s n-1 , s n ) with s n ∈ I ∩ N where

I = [σ n µ n/2 V n , (1 + ǫ)σ n µ n/2 V n ] = [σ n µ n/2 V n , σ n µ n/2 V n ] .
Since our computations rely on strict inequalities involving volume-computations which are continuous in ∆n-1 , this will imply the weakened form (without the factor l|k µ(l) l n-1 ) of Theorem 3.3. For k = 1, 2, . . . ∈ N we define finite subsets

I k = {s ∈ I ∩ N| n-1 i=0 s i x i = ks for some (x 0 , . . . , x n-1 ) ∈ B < √ µ-k 2 ∩ Z n } of natural integers in I ∩ N where B < √ µ-k 2 ∩ Z n
denotes the set of all integral vectors (x 0 , . . . , x n-1 ) ∈ Z n having (squared Euclidean) norm strictly smaller than µk 2 .

We have

| n-1 i=0 s i x i | ≤ n-1 i=0 s 2 i n-1 i=0 x 2 i ≤ µ→∞ µ n-1 V 2 n-1 4 n-1 ∆2 n-1 µ -k 2 ∼ µ→∞ 2 1-n µ n/2 V n-1 ∆n-1 for (x 0 , . . . , x n-1 ) ∈ B < √ µ-k 2 . This shows I k = {∅} if k ≥ A + 1 > 2 1-n V n-1 V n ∆n-1 σn .
An extension (s 0 , . . . , s n-1 , s n ) with s n ∈ I of a µ-sequence (s 0 , . . . , s n-1 ) is a µ-sequence if and only if s n ∈ A k=1 I k . Introducing the sets

X k (a) = {(x 0 , . . . , x n-1 ) ∈ Z n | n-1 i=0 s i x i ∈ kI ∩ (kN + a), n-1 i=0 x 2 i < µ -k 2 },
we have obviously ♯(I k ) ≤ ♯(X k (0)). This ensures the existence of a µ-sequence (s 0 , . . . , s n-1 , s n ) with

s n ∈ I ∩ N if we have A k=1 ♯(X k (0)) < ♯{I ∩ N}. (1)
Denoting by

X k ( * ) = {(x 0 , . . . , x n-1 ) ∈ Z n | 1 k n-1 i=0 s i x i ∈ I, n-1 i=0 x 2 i < µ -k 2 }
the union of the disjoint sets X k (0), X k (1), . . . , X k (k -1), the following asymptotic equalities hold.

Lemma 5.1 We have

♯(X k (j)) ∼ µ→∞ 1 k ♯(X k ( * ))
for j = 0, . . . , k -1.

It is thus enough to compute ♯(X k ( * )) in order to get an asymptotic estimation of

X k (0) ∼ µ→∞ 1 k ♯(X k ( * )). We have ♯(X k ( * )) = ♯{(x 0 , . . . , x n-1 ) ∈ Z n | 1 k n-1 i=0 s i x i ∈ I, n-1 i=0 x 2 i < µ -k 2 } ∼ µ→∞ Vol{(t 0 , . . . , t n-1 ) ∈ E n | n-1 i=0 t 2 i ≤ µ, 1 k n-1 i=0 s i t i ∈ I} and the requirement 1 k n-1 i=0 s i t i ∈ I amounts to the inequalities kσ n µ n/2 V n ≤ s i t i ≤ kσ n µ n/2 V n .
For huge µ (and fixed k), the number k♯(X k ) is thus essentially the volume W k of a subset of the n-dimensional ball of radius √ µ. More precisely, this subset is delimited by the two parallel affine hyperplanes orthogonal to (s 0 , . . . , s n-1 ) which are at distance

k D = k σn µ n/2 V n n i=0 s 2 i ∼ µ→∞ k √ µ 2 n-1 V n V n-1 ∆n-1 σn
and (1 + ǫ)k D of the origin.

We have thus

W k = k(1+ǫ) D k D µ -t 2 (n-1)/2 dtV n-1 ≤ ǫk D µ -k 2 D2 (n-1)/2 V n-1 ≤ µ→∞ ǫkσ n µ n/2 2 n-1 V n ∆n-1 1 -k 2 2 n-1 σn Vn V n-1 ∆n-1 2 (n-1)/2 . Using the asymptotic equalities ♯(X k ) ∼ µ→∞ W k k , we get A k=1 ♯(X k ) ≤ µ→∞ ǫσ n µ n/2 2 n-1 ∆n-1 V n A k=1 1 -k 2 2 n-1 σn ∆n-1 V n V n-1 2 (n-1)/2
.

Together with the obvious estimation

♯{I ∩ N} ∼ µ→∞ ǫσ n µ n/2 V n ,
we have now

♯{I ∩ N} A k=1 ♯(I k ) ≥ µ→∞ 2 1-n ∆n-1 A k=1 1 -k 2 2 n-1 σn ∆n-1 Vn V n-1 2 (n-1)/2 > 1
by assumption on the choice of σn . This proves the weak version (without the factor l|k µ(l)

l n-1 ) of Theorem 3.3 by inequation (1) since ♯{I ∩ N} -→ ∞ if µ → ∞.
We consider now intersections among the sets I 1 , I 2 , . . . , I A in order to deal with the factor l|k µ(l) l n-1 . This leads to a slightly better estimation of ♯( A k=1 I k ) and completes the proof of Theorem 3.3.

Call an element x = (x 0 , . . . , x n-1 ) ∈ X k (0) primitive if it is not of the form hZ n for an integral divisor h > 1 of k. Call x imprimitive otherwise. An imprimitive element is of the form hx with x ∈ X k/h (0) and contributes a common integer to the sets I k and I k/h . This implies the inequality

♯( A k=1 I k ) ≤ A k=1 ♯(X k (0) p )
where X k (0) p ⊂ X k (0) denotes the set of all primitive elements in X k (0).

It is thus enough to estimate the number of primitive elements in X k (0). We have

♯(X k ( * ) ∩ hZ n ) ∼ µ→∞ 1 h n ♯(X k ( * )). We have obviously X k (a) ∩ hZ n = ∅ for a ∈ hZ. Applying Lemma 5.1, obviously modified, to the sublattice hZ n ⊂ Z n of index h n shows ♯(X k (αh) ∩ hZ n ) ∼ µ→∞ 1 k/h ♯(X k ( * ) ∩ hZ n ) for α = 0, 1, . . . , k h -1. We get thus ♯(X k (0) ∩ hZ n ) ∼ µ→∞ 1 kh n-1 ♯(X k ( * )) ∼ µ→∞ 1 h n-1 ♯(X k (0)).
Since an element x ∈ X k (0)∩hZ n belongs also to X k (0)∩lZ n for any natural divisor l of h and since l|h µ(l) = 0 for h ≥ 2, the number ♯(X k (0) p ) of primitive elements in X k (0) is asymptotically given by

  l|k µ(l) l n-1   ♯(X k (0)).

This leads to the majoration

♯( A k=0 I k ) ≤ µ→∞ A k=1   l|k µ(l) l n-1   ♯(X k (0))
and proves Theorem 3.3.

2 Proof of Lemma 5.1 The statement of Lemma 5.1 is equivalent to the asymptotic equalities

♯(X k (j)) -♯(X k (i)) ♯(X k ( * )) ∼ µ→∞ 0 for 0 ≤ i, j < k. Fix 0 ≤ i < j < k.
Associate to an element (x 0 , x 1 , . . . , x n-1 ) ∈ X k (j) the element (x 0 + ij, x 1 , . . . , x n-1 ) provided that it belongs to X k (i). This induces a bijection between subsets Xk (j) and Xk (i) of X k (j), X k (i). The set of "bad" points

B k (i, j) = X k (j) \ Xk (j) ∪ X k (i) \ Xk (i)
consists of some integral points at bounded Euclidean distance < k ≤ A from the boundary ∂Z k of the the set

Z k = {(z 0 , . . . , z n-1 ) ∈ R n | n-1 i=0 s i z i ∈ kI, n-1 i=0 z 2 i ≤ µ -k 2 } .
This shows that

|♯(X k (j)) -♯(X k (i))| ≤ ♯(B k (i, j)) ≤ vol N k+ √ n/2 (∂Z k ) ∼ O(µ n-1 )
where

N k+ √ n/2 (∂Z k ) ⊂ R n denotes the set of all points at distance ≤ k + √ n/2 from the boundary ∂Z k of Z k . Since ♯(X k ( * )) = O(µ n ) this proves Lemma 5.1. 2
6 Proof of Theorem 1.5

Although an intuitively correct proof using Theorem 1.1 of Theorem 1.5 is easy, a rigorous proof is somewhat tedious and is the content of this section.

Two auxiliary results

The main ingredient for proving Theorem 1.5 is the following result which says essentially that an attracting fixpoint of a dynamical system is structurally stable.

Proposition 6.1 Given a real interval A ⊂ R, let f 1 , f 2 , . . . : A -→ A be a sequence of uniformly converging functions with continuous and differentiable limit f (x) = lim n→∞ f n (x) on A. Suppose that f has a fixpoint ξ = f (ξ) ∈ A and suppose that we have sup

x∈A |f ′ (x)| = λ < 1.
Then the sequence s n (x) of functions defined recursively by s 0 (x) = x and s n (x) = f n (s n-1 (x)) converges pointwise to the constant function ξ.

Proof Given δ > 0 there exists an integer N such that |f n (x)f (x)| < δ(1λ) for all x ∈ A and for all n > N . We have then for m > N

|s m (x) -ξ| = |f m (s m-1 (x)) -ξ| < δ(1 -λ) + |f (s m-1 (x)) -ξ| < δ -λδ + λ|s m-1 (x) -ξ| < δ -λ 2 δ + λ 2 |s m-2 (x) -ξ| . . . < δ -λ m-N δ + λ m-N |s N (x) -ξ| .
This shows

|s m (x) -ξ| < 2δ if m > max(N, N + log δ |s N (x) -ξ| /log(λ))
and implies the result since we can choose δ > 0 arbitrarily small. 2 Remark 6.2 (i) The proof of Proposition 6.1 shows in fact (ii) If the sequence f n (x) = F (x, 1/n) satisfies the hypotheses of Proposition 6.1 with F (x, y) having continuous partial derivatives of all orders up to k + 1 in a neighbourhood of (ξ, 0), then there exist constants a 1 , a 2 , . . . , a k such that

|s n (x) -ξ| ≤ λ n |x -ξ| + n k=1 λ n-k sup x∈A |f k (x) -f (x)| .
s n (x) = ξ + 1 1 -∂/∂ x F   k j=1 a j j! n -j   + O(n -(k+1) )
where ∂ a+b F ∂x a ∂y b denotes the obvious partial derivative of F (x, y), evaluated at (ξ, 0). The formulae for the first three coefficients a 1 , a 2 , a 3 are

a 1 = ∂/∂ y F a 2 = 2a 1 + (a 1 ∂/∂ x + ∂/∂ y ) 2 F a 3 = 12a 2 -6a 1 + 6∂/∂ y (a 1 ∂/∂ x + ∂/∂ y ) F + (a 1 ∂/∂ x + ∂/∂ y ) 3 F
In particular, for F (x, y) analytic and non-constant in y, the sequence s n (x) is asymptotically independent from x.

For x ∈ (0, ∞) we consider the real analytic positive function

τ (x) = ∞ k=1 e -π(k/x) 2 = 1 2 θ 3 ( i x 2 ) - 1 2 ,
related to the third Jacobi-theta function θ 3 (z) = k∈Z e iπk 2 z , cf. for instance Equation ( 6), page 102 in [START_REF] Conway | Sphere packings, Lattices and Groups[END_REF]. For x > 0, we have τ ′ (x) = 2π x 3 ∞ k=1 k 2 e -π(k/x) 2 > 0 and the easy inequalities

x 2 -1 < ∞ 0 e -π x 2 t 2 dt - 1 0 e -π x 2 t 2 dt < ∞ k=1 e -π(k/x) 2 < ∞ 0 e -π x 2 t 2 dt = x 2
for x > 0 imply that x -→ τ (x) is an increasing analytic diffeomorphism of (0, ∞). The equation

1 x = τ Ω(x) x = ∞ k=1 e -k 2 π(x/Ω(x)) 2
defines thus a real positive analytic function Ω : (0, ∞) -→ R. Equivalently, the function Ω is given by Ω(x) = xψ 1 x where the analytic diffeomorphism ψ satisfies ψ(τ (x)) = τ (ψ(x)) = x for all x > 0 and is the reciprocal function of τ .

The proof of Theorem 1.5 uses the following result.

Ω ′ (ξ) = 1 - τ (1) τ ′ (1) = 1 - ∞ k=1 e -πk 2 2π ∞ k=1 k 2 e -πk 2 ∼ 0.9135652 < 1 .

Proof of Theorem 1.5

Given an (n -1)-dimensional lattice of density ∆n-1 , Theorem 3.3 implies the existence of an n-dimensional lattice with density ∆n arbitrarily close to 1 2 n σ for σ > 0 defined by

2 n-1 ∆n-1 A k=1 1 -k 2 2 n-1 ∆n-1 V n V n-1 σ 2 n-1 = 1 20 where A = 2 1-n V n-1 V n ∆n-1 σ .
Given a positive constant ǫ > 0 and a natural integer N ∈ N, there exists thus a sequence of lattices Λ

d n-1 An k=1 1 -k 2 d n-1 d n V n V n-1 2 n-1 = 1 with A n = d n V n-1 d n-1 V n .
Equivalently, the sequence d 1 , d 2 , . . . is given by d

1 = 2, d 2 = f 1 (2), d 3 = f 2 (d 2 ), . . . , d n+1 = f n (d n ), .
. . where f 1 , f 2 , . . . : (0, ∞) -→ (0, ∞) are the functions defined implicitely by the equations

x ⌊fn(x)Vn/(xV n+1 )⌋ k=0 1 -k 2 xV n+1 f n (x)V n 2 n = 1 . Stirlings formula n! = √ 2πn(n/e) n (1 + O(1/n)) shows V n+1 /V n = √ π (n/2)! ((n + 1)/2)! = 2π/n (1 + O(1/n)) .
We have thus asymptotically

1 = x ⌊fn(x)Vn/(xV n-1 )⌋ k=0 1 -k 2 xV n+1 f n (x)V n 2 n = x ∞ k=1 e -k 2 π(x/fn(x)) 2 (1 + O(1/n))
and f n (x) -→ Ω(x) uniformly on any compact subset of (0, ∞). By Proposition 6.3 we can find The first column shows the indices n, choosen as successive powers of 2. The second column shows the corresponding value of d n . The third column shows the (n-1)-th iteration of Ω, starting from the initial value 2. The last column is the difference between the second and third column, multiplied by n and illustrates the expected finer asymptotic properties.

α < ξ = ∞ k=1 e -πk 2 -1 ∼ 23.14 < β such that Ω ′ (x) ≤ 19/20 for x ∈ [α, β]. We have thus uniform convergency f n (x) -→ Ω(x) for x ∈ [α, β],
Asymptotically This implies that Ω restricts to an increasing diffeomorphism from (2, ∞) onto (Ω(2), ∞) and since Ω(x) > 2, the map x -→ Ω(x) has a unique fixpoint at ξ = 1 τ (1) . 2

Final remarks

The inequality

♯( A k=1 I k ) ≤ A k=1 ♯(X k (0) p )
appearing in the proof of Theorem 3.3 is probably not sharp. A smaller upper bound for the cardinality ♯( A k=1 I k ) would thus improve the results of this paper.

The inequality above can be decomposed into the two inequalities for {I k 1 ,p , . . . , I k l ,p } ⊂ {I 1,p , . . . , I A,p } a subset of l distinct elements, one can neglect the contributions corresponding to k = 2, . . . , A. This would lead to a small improvement. A probably much more important improvement would result from a better understanding of the inequality ♯(I k,p ) ≤ ♯(X k (0) p ).

Instead of working with sublattices of Z n+1 orthogonal to a given vector (s 0 , . . . , s n ) ∈ Z n+1 , it is possible to consider sublattices Z n+a which are orthogonal to a set of a ≥ 2 linearly independent vectors in Z n+a . One might also replace the standard lattice Z n+1 by other lattices, e.g. sublattices of dimension n in Z n+1 (which approximate homothetically an arbitrary lattice by Proposition 2.2) or of finite index in Z n+1 .

Extending finite µ-sequences in an optimal way into longer µ-sequences amounts geometrically to the familiar process of lamination for lattices (see for instance [START_REF] Conway | Sphere packings, Lattices and Groups[END_REF] or [START_REF] Martinet | Perfect Lattices in Euclidean Spaces[END_REF]). The existence of an integer s ∈ I \ I 1 implies indeed the existence of a point P ∈ E n-1 which is far away from any lattice point of the affine lattice {(x 0 , . . . , x n-1 ) | x i s i = s} ⊂ Z n and corresponds thus to a "hole" of the lattice.

Proposition 2 . 2

 22 The set L n is dense in the set of similarity classes of n-dimensional Euclidean lattices. There are thus lattices in L n with densities arbitrarily close to the maximal density ∆ n of n-dimensional lattices. Proof of Proposition 2.2 Given a Gram matrix G = b i , b j of an n-dimensional lattice Λ = ⊕ n j=1 Zb j , Gram-Schmidt orthogonalization of the Z-basis b 1 , . . . , b n (with respect to the Euclidean scalar product) yields a matrix factorization G = L L t

Asymptotically, we have

  thus |s n (ξ) -ξ| = O(sup x∈N (ξ) |f n (x)f (x)|) (where N (ξ) ⊂ A is an arbitrarily small fixed neighbourhood of ξ) if sup x∈N (ξ) |f n (x)f (x)| is decreasing at a slower rate than powers of λ.

Proposition 6 . 3 1 ∞

 631 The applicationx -→ Ω(x) = xψ 1 xdefines a continuous map from (0, ∞) onto (2, ∞) which is strictly increasing for x > 2. It has a unique fixpoint ξ = 1 τ (1) = k=0 e -πk 2 ∼ 23.13882534 which is attracting under iteration since

6. 3 e -π(k 2 - 1 )/Y 2 - 1 =2πk 2 - 2 which 1 ∞

 32121221 Proof of Proposition 6.3Using the orientation-reversing diffeomorphismx = 1 τ (Y ) -→ Y = ψ 1 x of (0, ∞) we have Y τ (Y ) = Ω(x) = Ω 1 τ (Y ).The inequality τ (Y ) < Y 2 shows Ω(x) = Y τ (Y ) > 2 and 2Y Y -2 > Y τ (Y ) implies lim x→0 + Ω(x) = 2. Since lim Y →0 + Y τ (Y ) = lim Y →0 + Y e π 2 /Y 2 1 + ∞ k=2 ∞ the map Ω is a surjection onto (2, ∞). Since x -→ Y is orientation reversing, ddx Ω(x) > 0 is equivalent to strict positivity of Y 2 e -π(k/Y ) k=1 e -πk 2 /2 ∼ 1.38 .

  k,p ) and ♯(I k,p ) ≤ ♯(X k (0) p ) where we denote by I k,p ⊂ I k the subset of integers corresponding to primitive elements. If the subsets I 1,p , . . . , I A,p are asymptotically "independent" in the sense that ♯( l j=1 I k j ,p )/♯(I ∩ Z) ∼ µ→∞ l j=1 (♯(I k j ,p )/♯(I ∩ Z)) ,

  1 , Λ 2 , . . . , Λ N of dimensions 1, 2, . . . , N with densities ∆1 = 1, ∆2 , . . . , ∆N satisfying ∆m ≥ (1ǫ) d m 2 m , m = 1, . . . , N where d 1 = 2 and d 2 , d 3 , . . . , d N are recursively defined by the equation

  and there exists an integer N ξ such that f n ([α, β]) ⊂ [α, β] for all n ≥ N ξ . Proposition 6.1 shows now lim n→∞ d n = ξ which ends the proof. 2 The following Table illustrates the convergence of the sequence d 1 = 2, d 2 = f 1 (d 1 ), . . .:

	1 2.00000000 2.00000000	0
	2 3.62759873 3.99997210 -0.7447467 4 8.08369319 7.92472241 0.6358831
	8 18.71971890 14.38756801 34.6572071
	16 30.69030131 20.71395996 159.6214617
	32 29.45114255 22.98242063 206.9991014
	64 25.53248635 23.13821340 153.2334688
	128 24.17810739 23.13882533 133.0281029
	256 23.63011883 23.13882534 125.7711333
	512 23.37820694 23.13882534 122.5633803
	1024 23.25703467 23.13882534 121.0463495

  , the number d n is roughly given by

	23.13882534 + 119.58193	1 n	+ 1473.8282	1 n 2 + 25774.448	1 n 3 + . . .

(cf assertion (ii) of Remark 6.2).

(n-1)/2 < 1

The present version of this paper ows much to Fedor Petrov whose pertinent questions clarified and improved (and changed the title of) a preliminary version, see [1].

I thank also A. Marin, J. Martinet, P. Sarnak, B. Venkov and J-L. Verger-Gaugry for helpful comments and interest in this work.

* Support from the Swiss National Science Foundation is gratefully acknowledged.