SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N)

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1 SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) S.BOUCKSOM Abstract. The goal of ths note s to present a remarably smple proof, due to Hen, of a result prevously obtaned by Gllet-Soulé, relatng the number of lattce ponts n a symmetrc convex body to ts successve mnma. Introducton Let be a norm on R N, and denote by B ts closed unt ball (whch can thus be any symmetrc convex body). Let also M(B) := # ( B Z N) be the number of lattce ponts n B. Recall that the successve mnma of the lattce Λ := Z N wth respect to the norm are defned as The man result we am at s: r = r (Λ, B) = nf {r > 0 r (rb Λ) }. Theorem A. Wth the above notaton we have 1 N! M(B) r 6 N. As we shall see, the lower bound s a straghtforward consequence of a wellnown varant, due to Van der Corput, of Mnows s frst theorem. The upper bound s bascally [Hen02, Theorem 1.5]. A result of ths nd s stated (n a less explct way) n [GS91, Proposton 6]. Among other thngs, the proof provded there uses Mnows s second theorem and the dffcult Bourgan-Mlman theorem. We note that n the applcatons to the study of the arthmetc volume gven n [BC11], only the coarser estmate r <1 log M(B) = r <1 log r + O (N log N) s actually used. Date: Aprl 16,

2 2 S.BOUCKSOM 1. Mnows s frst theorem The followng varant of Mnows s frst theorem s due to Van der Corput. Proposton 1.1. We have M(B) 2 N vol(b). Tang to (1 + ε) tmes the l -norm shows that the constant 2 N cannot be mproved. Lemma 1.2 (Blchtfeld s prncple). Let A R N be a measurable set wth fnte volume. Then some translate of A contans at least vol(a) lattce ponts. Proof. Set m := vol(a) 1. Let P be a fundamental doman for Λ, so that vol(p ) = 1. For each v Λ set and consder A v := (A (P + v)) v, f := v Λ 1 Av. Snce each A v s contaned n P, we have sup f(x) f(x)dx = vol(a v ) x P P v Λ = vol (A (P + v)) = vol(a) > m. v Z N It follows that there exsts x P belongng to at least m + 1 A v s, hence the result. Proof of Proposton 1.1. Set m := 2 N vol(b) 1 and A := 2 1 B. Snce vol (A) > m, Lemma 1.2 shows that A contans m + 1 dstnct ponts v 0,..., v m such that v v j Λ for all, j. It follows that v 1 v 0,..., v m v 0 are m dstnct and non-zero lattce ponts n B, and we get as desred M(B) m Hen s theorem The followng result wll mmedately mply the upper bound n Theorem A. Theorem 2.1. [Hen02, Theorem 1.5] We have M(B) 2 N We start wth two easy lemmas. N 2r Lemma 2.2. There exsts a Z-bass (e ) of Λ such that for each = 1,..., N. Λ r B Ze Ze 1 We then say that (e ) s an adapted bass of Λ (wth respect to B).

3 SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) 3 Proof. By compactness, we may fnd lnearly ndependent vectors u 1,..., u N Λ such that u = r. It s then easy to chec that r = mn { u u Λ \ (Ru Ru 1 )}. (2.1) We then choose a Z-bass (e 1,..., e N ) of Λ such that and the result follows easly. u Ze Ze, Lemma 2.3. Gven a non-ncreasng sequence a 1... a N of postve ntegers, there exsts postve ntegers b 1,..., b N such that () a b < 2a for all ; () b +1 dvdes b for all. Proof. Set b := 2 m a N wth m := mn{m N 2 m a N a }. Proof of Theorem 2.1. For each set a = 2r 1 + 1, so that a s the smallest postve nteger wth 2a 1 < r. Pc b 1,..., b N as n Lemma 2.3, let (e ) be an adapted bass of Λ as n Lemma 2.2, and consder the sublattce of fnte ndex Λ := Zb e. We clam that Λ 2B = {0}. Indeed, let u Λ wth u 2, and wrte u = m b e wth m Z. If u s non zero, set := max { m 0}. Snce b dvdes b 1,..., b 1, we then have b 1 u Λ, and u 2b 1 2a 1 < r. b 1 By defnton of an adapted bass, we should thus have b 1 u Ze Ze 1, a contradcton. As a consequence of the clam, the projecton map π : R N R N /Λ s njectve on B, so that M(B) = #π(b Λ). But π(b Λ) s contaned n Λ/Λ, and hence M(B) = #π(b Λ) [Λ : Λ ] = b 2 N a. 3. Proof of Theorem A Let r 1... r n < 1, 1 n N, be those successve mnma that are less than 1. To prove the lower bound, pc lnearly ndependent vectors u 1,..., u n Λ such that u = r, and ntroduce the hypercube P := Conv{±r 1 u 1 r}

4 4 S.BOUCKSOM and the dscrete abelan group Λ := Zu Zu n. Snce P s contaned n B, we then have by Proposton 1.1 M(B) #(P Λ ) 2 n vol Λ (P ), where vol Λ denotes the Haar measure on Ru Ru n normalzed by ts lattce Λ. Snce vol Λ (P ) = 2n n! (r 1... r n ) 1, the lower bound n Theorem A follows. We now turn to the upper bound. We have 2r for > n. We thus get and 2r 1 N 2r N r <1 r r 1 for = 1,..., n, and the upper bound n Theorem A s now a consequence of Theorem Mnows s second theorem As a drect consequence of Theorem 2.1, we get the followng verson of Mnows s second theorem: 2 N N N! vol(b) r 4 N. (4.1) Indeed, the upper bound follows drectly from Theorem 2.1, replacng B wth tb and lettng t +, usng the obvous scalng property r (Λ, tb) = t 1 r (Λ, B) for t > 0. As to the lower bound, t s obtaned as n the proof of Theorem A: the polytope P s contaned n B, and hence vol(b) vol(p ) = 2N det ( r1 1 N! u 1,..., r 1 N u N), whch yelds the lower bound follows snce det (u 1,..., u N ) s a non-zero nteger. For comparson, the classcal statement of Mnows s second theorem s as follows: Theorem 4.1. We have 2 N N N! vol(b) r 2 N. It thus mproves on the upper bound n (4.1) by a factor 2 N. We now present Esterman s proof of Theorem 4.1, followng [GL87, pp.58-61]. Proof of Theorem 4.1. We have already recalled the standard argument provng the lower bound. To get the upper bound, let (e 1,..., e N ) be an adapted bass of Λ as n Lemma 2.2, and set Λ 0 := {0} and Λ := Ze Ze

5 SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) 5 for = 1,..., N. Let ρ := V/Λ 1 V/Λ and π : V V/Λ be the natural projectons, and for each measurable subset S of V let v (S) := vol (π (S)), where vol denotes the normalzed Haar measure on V/Λ. By (2.1), f u s a lattce( pont ) such that u < r then u Λ 1. It follows that ρ s njectve on π r 1 2 B, so that ( r v ) ( r = v 1 ) for = 1,..., N, where we have set v 0 := vol and r 0 := 1. By Lemma 4.2 below ths mples hence ( rn 1 v N ) whch concludes the proof. ( r v ) ( ) N +1 r ( r 1 v 1 r 1 ), N ( r r 1 Lemma 4.2. Let C V be a convex subset. Then for all t 1. ) N +1 ( ) 1 v 0 = r 1... r N 2 N vol(b) v (tc) t N v (C) Proof. By translaton nvarance of the Haar measure on V/Λ, we may assume that 0 C. If we let V V be the vector space generated by Λ, then each fber F of τ : V/Λ V /Λ s canoncally somorphc to V/V, and vol(tc F ) = t N vol(c F ). Snce tc contans C, we have on the other hand vol(τ (tc)) vol(τ (C)), and the result follows by Fubn s theorem. References [BHW93] [BC11] [GS91] [GL87] [Hen02] U. Bete, M. Hen, J.M. Wlls. Successve-mnma-type nequaltes. Dscrete Comput. Geom. 9 (1993), S. Boucsom, H. Chen. Oounov bodes of fltered lnear seres. Compos. Math. 147 (2011), no. 4, H. Gllet, C. Soulé. On the number of lattce ponts n convex symmetrc bodes and ther duals. Israel J. Math. 74 (1991), no. 2-3, P. M. Gruber, C. G. Leererer. Geometry of numbers. Second edton. North- Holland Mathematcal Lbrary, 37. North-Holland Publshng Co., Amsterdam, M. Hen. Successve mnma and lattce ponts. IVth Internatonal Conference n Stochastc Geometry, Convex Bodes, Emprcal Measures and Applcatons to Engneerng Scence, Vol. I (Tropea, 2001). Rend. Crc. Mat. Palermo (2) Suppl. No. 70, part I (2002),

6 6 S.BOUCKSOM CNRS-Unversté Perre et Mare Cure, Insttut de Mathématques de Jusseu, F Pars Cedex 05, France E-mal address: