on distinct sums and distinct distances

Download On distinct sums and distinct distances

Post on 02-Jul-2016

213 views

Category:

Documents

0 download

Embed Size (px)

TRANSCRIPT

  • http://www.elsevier.com/locate/aimAdvances in Mathematics 180 (2003) 275289

    On distinct sums and distinct distances

    Gabor Tardos*

    Renyi Institute, Realtanoda utca 13-15, 1053 Budapest, Hungary

    Received 28 August 2001; accepted 2 August 2002

    Communicated by Laszlo Lovasz

    Abstract

    The paper (Discrete Comput. Geom. 25 (2001) 629) of Solymosi and Toth implicitly raised

    the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s2nsums of pairs of elements of the same set. What is the minimum number of distinct sums one

    can get this way? This paper proves that the number of distinct sums is at least nds ; where

    ds 1=cJs=2n is dened in the paper and tends to e1 as s goes to innity. Here e is the base ofthe natural logarithm. As an application we improve the SolymosiToth bound on an old

    Erd +os problem: we prove that n distinct points in the plane determine On 4e5e1e distinctdistances, where e40 is arbitrary. Our bound also nds applications in other related results indiscrete geometry. Our bounds are proven through an involved calculation of entropies of

    several random variables.

    r 2003 Elsevier Science (USA). All rights reserved.

    MSC: 52C10; 11B75

    Keywords: Distinct distances; Entropy; Erd +os problems

    1. Introduction

    For an n by s matrix A aij we dene SA faij aikj1pipn; 1pjokpsg theset of pairwise sums of entries from the same row. Let fsn be the minimum sizejSAj for a real n by s matrix with all its sn entries being pairwise distinct.The goal of this paper is to study the asymptotic behavior of fsn; especially for

    large constant values of s:

    ARTICLE IN PRESS

    *Fax: +36-14-83-83-33.

    E-mail address: tardos@renyi.hu.

    0001-8708/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

    doi:10.1016/S0001-8708(03)00004-5

  • The motivation for this problem comes from the breakthrough paper of Solymosi

    and Toth [10]. They proved an On6=7 bound for an old problem of Erd +os [4], theminimum number distinct distances n points determine in the plane. This resultsubstantially improved earlier works of Moser [6], Chung [2], Chung et al. [3], andSzekely [12]. See [7] for the background of this intriguing old Erd +os problem and forfurther references.

    Solymosi and Toth implicitly use f3n On1=3 in their proof, and a closer lookreveals that any stronger bound for fsn; with a constant s would improve theirresult. Section 4 has the details; Corollary 15 states the bound we get on the numberof distinct distances in the plane.

    We have that f2n 1; f3n Yn1=3 and f4n Yn1=3 but for f5n andabove the correct order of magnitude is unknown. The best current bounds for f5and f6 are

    n4=11pf5npf6n On2=5:

    These bounds are special cases of a construction of Ruzsa and Theorem 1 below.This special case of Theorem 1 (with a worse constant factor) has a much simplerproof than the full theorem as shown in Section 5.The best upper bound on fsn; i.e., the best construction is due to Ruzsa [9], he

    proves

    fsn On12 12s2

    for even s: The lower bound of the following theorem is stated for odd values of s: Itis interesting to note that both the known lower and upper bounds are identical forthe functions f2k1npf2kn but for kX3 we have no other indication for thesefunctions being close to each other.

    Theorem 1. For an integer kX2 we have

    f2k1nXn1ck ;

    where for kp14 we have

    ck Xki0

    1

    i! 1k 1k!;

    while for kX14 we have

    ck Xki0

    1

    i! k

    3 7k2 20k 40k4 8k3 26k2 46k 40k!:

    Notice, that both denitions of ck give the same value for c14:

    ARTICLE IN PRESSG. Tardos / Advances in Mathematics 180 (2003) 275289276

  • It is easy to see, that the limit of the values ck as k goes to innity is e; the base ofthe natural logarithm. Thus we have the following.

    Corollary 2. For every e40 we have a positive integer s se with

    fsnXn1=ee:

    Note, that the limit of the exponent in the Ruzsa construction is 12; so the lower and

    upper bounds are far apart.In Sections 2 and 3 we give the proof of Theorem 1. In Section 4, we apply it (or

    rather Corollary 2) to get an improvement over the SolymosiToth bound on thenumber of distinct distances n point determine in the plane (see Corollary 15). Wealso give references to other related problems where Corollary 2 could be used indiscrete geometry. In Section 5, we give an elementary and simple proof of the rst

    non-trivial case of Theorem 1: we prove that f5n On4=11: We close the paperwith concluding remarks and open problems in Section 6.

    2. The proofreduction to a linear program

    Let us x the positive integers s; n and an n by s real matrix A: Our proof does notuse in full generality the assumption that all entries of A are distinct. It is enough tomake the slightly weaker assumption that no two rows of A have two commonentries. Our goal is to prove a lower bound on jSAj:Let I f1; 2; 3;y; sg be the set of column indices. For subsets U ; VDI and for an

    s-tuple R a1;y; as we dene the UV pattern pU ;V R of R to be a sequence ofreal numbers consisting of the differences ai aj for i; jAU and for i; jAV and thesums ai aj for iAU and jAV : We dene

    HU ; V HpUV R;

    where H denotes the entropy and R is a uniformly distributed random row of A: Allentropies and all logarithms in this paper are binary.The next lemma stating linear constraints on the entropies HU ; V is crucial for

    the proof.

    Lemma 3. Let U ; U 0; V ; V 0DI : We have

    (a) HU ; V HV ; U;(b) HU ; VpHU 0; V 0 if UDU 0 and VDV 0;(c) HU ; V 0 if U | and jV j 1;(d) HU ; VplogjSAj if UaV and jU j jV j 1;(e) HU ; V log n if U-Va| and jU,V j41;(f) HU,U 0; V,V 0 HU-U 0; V-V 0pHU ; V HU 0; V 0 if

    U-U 0,V-V 0a|:

    ARTICLE IN PRESSG. Tardos / Advances in Mathematics 180 (2003) 275289 277

  • Proof. We use the well-known properties of the entropy to prove this lemma.

    A. Range: For a random variable F that has k possible values HFplog k withequality if F is distributed uniformly.Part (c) follows since pU ;V R is constant in that case.Part (d) also follows since pU ;V R consists of a single value from SA in that

    case.Part (e) of the lemma also follows from the above property. The pattern pU ;V R

    contains 2ai for the index iAU-V and thus it determines ai and with it all othervalues aj with jAU,V : As two entries uniquely determine the row of the matrix Awe have that pU ;V R is different for all the n rows of A; thus it is uniformlydistributed among n possible values.B. Monotonicity: If the value of a random variable F uniquely determines the

    value of another random variable G then we have HFXHG:Part (b) of the lemma follows as the pattern pU 0;V 0 R contains all entries of the

    pattern pU ;V R:Part (a) of the lemma also follows as the patterns pU ;V R and pV ;UR contain the

    same entries, so they mutually determine each other.C. Submodularity: Suppose that the value of either one of the random variables F1

    and F2 determines the value of the random variable G1 and the values of the randomvariables F1 and F2 together determine the value of the random variable G2: In thiscase, we have HG1 HG2pHF1 HF2:For part (f) of the lemma we use the submodularity of entropy as stated above.

    Clearly the pattern pU-U 0;V-V 0 R is determined by either one of pU ;V R andpU 0;V 0 R: We need to show an entry ai7aj in pU,U 0;V,V 0 R is determined by thetwo patterns pU ;V R and pU 0;V 0 R: Indeed, the term ai7aj in the former pattern canbe expressed as a sum or difference of the terms ai7ak and aj7ak in the latterpatterns if kAU-U 0,V-V 0: &

    Lemma 3 contains linear constraints on the entropies HU ; V and logjSAj; thussolving them as a linear program provides a bound on jSAj: This is indeed theroute we will take. The rest of the proof of the lower bound of jSAj uses solelyLemma 3. We remark here that the linear program dened by Lemma 3 has a uniqueoptimal solution for all values of s except for s 27 or s 28 where the optimalsolutions are the convex combinations of two extremal optimal solutions.Our rst step is to use averaging to decrease the exponential number of variables

    to less than s2 of them.For integers i; jX0; 1pi jps we dene

    Hi;j 1 1log nn

    ini

    jXU ;V

    HU ; V;

    where the summation extends over all nini

    j pairs of disjoint subsets U and V of I

    with jU j i and jV j j: (We consider the values Hi;j to form a matrix H with some

    ARTICLE IN PRESSG. Tardos / Advances in Mathematics 180 (2003) 275289278

  • entries of this matrix missing. We will only use the values Hi;j satisfying 0pi; jpkand 1pi jp2k 1 where k Js=2n:)

    Lemma 4. For i; j non-negative integers with 1pi jps we have:(a) (symmetry) Hi;j Hj;i;(b) (monotonicity) Hi;jXHi1; j if i jps 1;(c) H0;1 1;(d) H1;1X1 logjSAj=log n;(e) (convexity) Hi1; j Hi1; jX2Hi;j if iX1 and 2pi jps 1;(f) Hi;jXHi1; j Hi;j1 if i jps 1:

    We could also state the non-negativity of these variables, but we will not use it.

    Proof. Parts (a)(d) of this lemma follow from the corresponding parts of Lemma 3by simple averaging.Part (e) follows from part (f) of Lemma 3, here the averaging is over the four-

    tuples of sets U ; V ; U 0; V 0DI satisfying jU j jU 0j i; jU-U 0j i 1;V V 0; jV j j and U,U 0-V |:Finally, for part (f) of this lemma consider two disjoint subsets U and V 0 of I (not

    both the empty set) and an index kAI\U,V 0: Applying Lemma 3(f) for U ;U 0 U,fkg; V V 0,fkg; and V 0 one gets

    HU ; V 0 HU 0; VpHU ; V HU 0; V 0:

    Here Lemma 3(e) applies and yields HU 0; V log n; thus, we haveHU ; V 0 log npHU ; V HU 0; V 0:

    Part (f) of the lemma follow