A Lower Bound on the Density of Sphere Packings via Graph Theory. Michael Krivelevich, Simon Litsyn, and Alexander Vardy.

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1 IMRN Iteratioal Mathematics Research Notices 004, No. 43 A Lower Boud o the Desity of Sphere Packigs via Graph Theory Michael Krivelevich, Simo Litsy, ad Alexader Vardy 1 Itroductio A sphere packig P i R is a collectio of oitersectig ope spheres of equal radii, ad its desity P is the fractio of space covered by their iteriors. Let = sup P, 1.1 P R where the supremum is take over all packigs i R. A celebrated theorem of Mikowski states that ζ/ 1 for all.siceζ = 1 + o1, the asymptotic behavior of the Mikowski boud [9] is give by Ω. Asymptotic improvemets of the Mikowski boud were obtaied by Rogers [10], Daveport ad Rogers [4], ad Ball [], all of them beig of the form c, where c>0is a absolute costat. The best curretly kow lower boud o is due to Ball [], who showed that there exist lattice packigs with desity at least 1 ζ. I this paper, we use results from graph theory to prove the followig theorem. Theorem 1.1. For all sufficietly large, there exists a sphere packig P R such that P Moreover, the spheres i P ca be described usig a determiistic procedure whose complexity is at most O γ log for a absolute costat γ. Received 9 February 004. Revisio received 8 March 004. Commuicated by Peter Sarak.

2 7 Michael Krivelevich et al. Although the costat c = 0.01 i 1. is ot as high as i the bouds of Rogers c = 0.74, Daveport ad Rogers c = 1.68, ad Ball c =, Theorem 1.1 still provides a improvemet upo the Mikowski boud by the same liear i factor. With some effort, the costat i Theorem 1.1 ca be icreased by at least a factor of 10.However, the mai merit of our proof is ot so much i the result itself, but rather i the approach we use. Our argumet is essetially differet from all those previously employed, icludig the costructios of Mikowski [9], Hlawka [5], Rogers [10], Daveport ad Rogers [4], ad Ball []. Istead of relyig directly o the geometry of umbers, we apply powerful graph-theoretic tools; specifically, we use lower bouds [1, 3] o the idepedece umber of locally sparse graphs. A similar approach has bee recetly used by Jiag ad Vardy [7] i a asymptotic improvemet of the classical Gilbert-Varshamov boud i codig theory. We fid it remarkable that all the existig approaches to the Mikowski problem, while beig very differet i ature, result i the same liear i improvemet over the classical Mikowski boud. At this stage, we do ot have a satisfactory explaatio of this rather puzzlig pheomeo; perhaps, it idicates a iheret difficulty i breakig the liear barrier. Graph-theoretic proof of the Mikowski boud First, we ie two cubes i R a smaller cube K 0 of side s ad a larger cube K 1 of side s + r. Specifically, { K 0 = x R : } xi s, { K 1 = x R : x i s } + r,.1 where r ad s are, so far, arbitrary fuctios of, except that we assume r,s Z. Next, ie a graph G as follows: the vertices of G are give by VG = Z K 0, ad {u, v} EG if ad oly if du, v <r, where d, is the Euclidea distace i R.Let I be a maximal idepedet set i G ; that is, I is such that every vertex of VG \Iis adjacet to at least oe vertex i I. By the iitio of EG, spheres of radius r about the poits of I do ot overlap. Moreover, all such spheres lie iside the cube K 1.SiceK 1 tiles R, we coclude that there exists a sphere packig P with desity P = I V r Vol K 1 = I V r s + r,.

3 Sphere Packig via Graph Theory 73 where V is the volume of a uit sphere i R.Letd deote the maximum degree of a vertex i G.Itiswellkowad obvious, for ay graph G that I V G s d + 1 = + 1 d Let S r deote the ope sphere of radius r about the origi 0 of R.Iftheratios /r is sufficietly large, as we will assume, the d + 1 is just the umber of poits of Z cotaied i S r. Thus we ca roughly estimate d + 1 as simply the volume V r of S r. Combiig this estimate with.,.3 ad takig s = r, we obtai V s r P V = r s + r r s 1 + o1..4 Alteratively, a precise boud o d ca be derived as follows. With each poit v Z, we associate the uit cube { Kv = x R : 1 <x i v i 1 }..5 Such cubes are fudametal domais of Z ; hece, they do ot itersect. The legth of the mai diagoal of Kv is, which implies that dx, v / for all x Kv. Let D = S r Z, so that d + 1 = D. It follows by the triagle iequality that if v D, the dx, 0 <r + / for all x Kv. Hece Kv S r +..6 v D Expressig the volume of v D Kv as D = d + 1, this implies that the maximum degree of a vertex i G is bouded by d + 1 V r +..7

4 74 Michael Krivelevich et al. Combiig.7 with. ad.3, the takig r = ad s = 4 say, proves that the desity of P is at least V s r P V s + r r + 1 = 1 + r 1 + s 4r = 1 + o1..8 Asymptotically,.8 coicides with the Mikowski boud o. Sice a maximal idepedet set i G ca be foud i time O VG ad VG = s + 1 = 4 + 1, the boud i.8 also reproduces the result of Litsy ad Tsfasma [8, Theorem 1]. 3 Asymptotic improvemet usig locally sparse graphs Next, we take I to be a idepedet set of maximum size i G ad cosider bouds o I = αg that are sharper tha the trivial boud i.3. LetT deote the umber of triagles i G. The it is kow see [1] ad [3, Lemma 15, page 96] that α V G G log 10d d 1 log T V G. 3.1 Now, let t be the smallest iteger with the property that for all v VG, the subgraph of G iduced by the eighborhood of v has at most t edges. The it follows from 3.1 that α G s d log d 1 log t Thus, to obtai a asymptotic improvemet upo the Mikowski boud, it would suffice to prove that t = od. Before divig ito the techical details of the proof, we explai ituitively why we expect to get a improvemet by a factor that is liear i. We pick two poits x ad y uiformly at radom i S r. The relevat questio is: what is the probability that dx, y <r? It is a rather stadard fact i highdimesioal geometry that regardless of the value of r this probability behaves as e c for large. Therefore, we should expect that oly a expoetially small fractio of pairs of poits of Z, lyig withi a sphere of radius r cetered at some z VG,

5 Sphere Packig via Graph Theory 75 are adjacet i G. I other words, we expect that t d /e c which, i view of 3., immediately leads to the desired Θ improvemet factor. We derive a rigorous upper boud o t ext. Cosider the eighborhood of 0 VG ad let H deote the subgraph of G iduced by this eighborhood. As i Sectio, we assume that the ratio s /r i.1 is sufficietly large so that VH = Z \{0} S r. It is the obvious that t = EH, so t = degx, 3.3 x VH where degx deotes the degree of x i H.WriteS 1 = S r ad let S be the sphere of radius r about x VH.The, degx is just the umber of poits of Z i S 1 S. Usig the same argumet as i.6, we thus have degx Vol S 1 S, 3.4 where S 1 = S r + / ad S is the sphere of radius r + / about x. Clearly, the right-had side of 3.4 depeds o x oly via its distace to the origi. Hece, ie ρ = r + ad δ x = dx, 0 ρ 3.5 with δ x 0, 1/ for all x. Itisotdifficult to write dow a precise expressio for the volume of S 1 S i terms of ρ ad δ x.letθ = cos 1 δ x.the, VolS 1 S is twice the volume of a spherical sector of agle θ shaded i Figure 3.1 mius twice the volume of a right coe of the same agle cross-hatched i Figure 3.1. Thus, Vol S 1 S ρ V 1 = 1 θ 0 si ϕ dϕ δ x si θ However, rather tha estimatig the itegral θ 0 si ϕ dϕ i 3.6, we will use the followig simple boud without compromisig much i the asymptotic quality of the obtaied result. It is easy to see cf. Figure 3. that S 1 S is cotaied i a cylider of height ρ dx, 0 whose base is a 1-dimesioal sphere of radius ρ si θ. Hece, Vol S 1 S ρ 1 δx V 1 ρ 1 si θ 1 1 δ x /V ρ, 3.7

6 76 Michael Krivelevich et al. ρ θ 0 x Figure 3.1 ρ ρsi θ θ 0 x Figure 3. where the secod iequality follows from the fact that V 1 V for all.now, let U k deote the set of all x VH such that d x, 0 = k. Clearly, d x, 0 is a iteger i the rage 1 d x, 0 4r 1. We thus break the sum i 3.3 ito two parts: t = x VH degx = r /4 1 k=1 x U k degx + 4r 1 k=r /4 x U k degx, 3.8 ad boud each part separately. As it turs out, crude upper bouds o U k suffice i each case. For the first double sum i 3.8, we use the fact that degx d V ρ for

7 Sphere Packig via Graph Theory 77 all x VH by.7. Therefore, applyig oce agai the method of.6, we have r /4 1 k=1 x U k degx V ρ r /4 1 k=1 Uk V ρ r + Vol S 1 V ρ, 3.9 where the last iequality assumes r /, so that ρ r +. I fact, heceforth, we take r = as i Sectio. Thek 4 i the secod sum of 3.8, ad we ca boud U k as follows: U k VolS k + / V k / 1 + / V k /. Combiig this with the bouds 3.4 ad 3.7 o degx, we have 4r 1 k=r /4 x U k degx V ρ 4r 1 k=r /4 4r 1 +1 Vρ U k 1 δ k / k=r /4 δ k 1 δ k /, 3.10 where δ k = k/ρ. Now, the fuctio fδ = δ 1 δ attais its maximum i the iterval [0, 1/] at δ = 1/. Puttig this together with 3.8 ad 3.9, we fially obtai the desired upper boud o t, amely, t +1 1 V ρ + Vρ 4r 1 k=r /4 / V ρ Recall that d V ρ by.7. Substitutig this boud together with 3.11 i the boud 3. o the idepedece umber of G produces α s + 1 G 10V ρ log V ρ 1 3 log V ρ 1 log ρ s + 1 log log r + 3 = 10V r +, 3.1

8 78 Michael Krivelevich et al. where we have used the iitio of ρ i 3.5.Fially, usig. with I = αg, while takig r = ad s = 4 as before, we obtai the followig boud: P 0 s s + r r + r log 3 log r + log 3 = 1 + o Sice log / 3/0 = , this establishes 1.. Now, give ay graph G = V, E, there is a determiistic algorithm [6] that fids a idepedet set I i G whose size is lower-bouded by 3.1 i time Od av E + V, where d av is the average degree of G.Ithe case of the graph G, this reduces to OVs + 1 r + /.Withr = ad s = 4, the expressio Vs +1 r + / behaves as 64πe 7 for large.this completes the proof of Theorem 1.1, with the value of γ give by 7 + ɛ.however, ote that ourchoiceofthevaluesr = ad s = 4 was motivated primarily by the otatioal coveiece of havig r,s Z. I fact r = 1.5+ɛ ad s =.5+ε would suffice, as ca be readily see from.8 ad Thus, the value of γ ca be take as ɛ. Ackowledgmet This work was supported i part by the US-Israel Biatioal Sciece Foudatio Michael Krivelevich, the Israel Sciece Foudatio Michael Krivelevich ad Simo Litsy, ad the Packard Foudatio ad the Natioal Sciece Foudatio Alexader Vardy. Refereces [1] M. Ajtai, J. Komlós, ad E. Szemerédi, A ote o Ramsey umbers, J. Combi. Theory Ser. A , o. 3, [] K. Ball, A lower boud for the optimal desity of lattice packigs, It. Math. Res. Not , o. 10, [3] B. Bollobás, Radom Graphs, Academic Press, Lodo, [4] H. Daveport ad C. A. Rogers, Hlawka s theorem i the geometry of umbers, Duke Math. J , [5] E. Hlawka, Zur geometrie der zahle, Math. Z , Germa.

9 Sphere Packig via Graph Theory 79 [6] T. Hofmeister ad H. Lefma, Idepedet sets i graphs with triagles, Iform. Process. Lett , o. 5, [7] T. Jiag ad A. Vardy, Asymptotic improvemet of the Gilbert-Varshamov boud o the size of biary codes, to appear i IEEE Tras. Iformatio Theory. [8] S. Litsy ad M. Tsfasma, Costructive high-dimesioal sphere packigs, Duke Math. J , o. 1, [9] H. Mikowski, Diskotiuitäsbereich für arithmetische Aequivallez, J. reie agew. Math , 0 74 Germa. [10] C. A. Rogers, Existece theorems i the geometry of umbers, A. of Math , Michael Krivelevich: Departmet of Mathematics, Tel Aviv Uiversity, Tel Aviv 69978, Israel address: krivelev@post.tau.ac.il Simo Litsy: Departmet of Electrical Egieerig-Systems, Tel Aviv Uiversity, Tel Aviv 69978, Israel address: litsy@eg.tau.ac.il Alexader Vardy: Departmet of Electrical Egieerig, Departmet of Computer Sciece, ad Departmet of Mathematics, Uiversity of Califoria, Sa Diego, 9500 Gilma Drive, La Jolla, CA 9093, USA address: vardy@kilimajaro.ucsd.edu