For infinitely many primes p = 4k+1 we give a slightly

improved upper bound for the maximal cardinality of a set B ⊂ Z

p

such that the difference set B−B contains only quadratic residues.

Namely, instead of the ”trivial” bound |B| ≤ √p we prove |B √p | ≤ − 1, under suitable conditions on p. The new bound is valid

for approximately three quarters of the primes p = 4k + 1

Bachoc, C.

Matolcsi, Máté

Ruzsa, Z. Imre

English

Repository of the Academy's Library

arXiv:1305.0577v1  [math.CO]  2 May 2013SQUARES AND DIFFERENCE SETS IN FINITEFIELDSC. BACHOC, M. MATOLCSI, AND I. Z. RUZSAAbstract. For infinitely many primes p = 4k+1 we give a slightlyimproved upper bound for the maximal cardinality of a set B ⊂ Zpsuch that the difference set B−B contains only quadratic residues.Namely, instead of the ”trivial” bound |B| ≤ √p we prove |B| ≤√p − 1, under suitable conditions on p. The new bound is validfor approximately three quarters of the primes p = 4k + 1.Keywords: quadratic residues, Paley graph, maximal cliques.MSC2010 classification: 05C69, 11T06.1. introductionLet q be a prime-power, say q = pk. We will be interested in es-timating the maximal cardinality s(q) of a set B ⊂ Fq such that thedifference set B −B contains only squares. While our main interest isin the case k = 1, we find it instructive to compare the situation fordifferent values of k.This problem makes sense only if −1 is a square; to ensure thiswe assume q ≡ 1 (mod 4). The universal upper bound s(q) ≤ √qcan be proved by a pigeonhole argument or by simple Fourier anlysis,and it has been re-discovered several times (see [7, Theorem 3.9], [11,Problem 13.13], [3, Proposition 4.7], [2, Chapter XIII, Theorem 14],[10, Theorem 31.3], [9, Proposition 4.5], [6, Section 2.8] for variousproofs). For even k we have equality, since Fpk can be constructed as aquadratic extension of Fpk/2, and then every element of the embeddedfield Fpk/2 will be a square. It is known that every case of equality canbe obtained by a linear transformation from this one, [1].Such problems and results are often formulated in terms of the Paleygraph Pq, which is the graph with vertex set Fq and an edge betweenx and y if and only if x− y = a2 for some non-zero a ∈ Fq.M.M. and I.Z.R. were supported by the ERC-AdG 228005, and OTKA GrantsNo. K81658, and M.M. also by the Bolyai Scholarship.12 C. BACHOC, M. MATOLCSI, AND I. Z. RUZSAPaley graphs are self-complementary, vertex and edge transitive, and(q, (q−1)/2, (q−5)/4, (q−1)/4)-strongly regular (see [2] for these andother basic properties of Pq). Paley graphs have received considerableattention over the past decades because they exhibit many propertiesof random graphs G(q, 1/2) where each edge is present with probability1/2. Indeed, Pq form a family of quasi-random graphs, as shown in [4].With this terminology s(q) is the clique number of Pq. The generallower bound s(q) ≥ (12+ o(1)) log2 q is established in [5], while it isproved in [8] that s(p) ≥ c log p log log log p for infinitely many primesp. The “trivial” upper bound s(p) ≤ √p is notoriously difficult toimprove, and it is mentioned explicitly in the selected list of problems[6]. The only improvement we are aware of concerns the special casep = n2 + 1 for which it is proved in [12] that s(p) ≤ n − 1 (the sameresult was proved independently by T. Sanders – unpublished, personalcommunication). It is more likely, heuristically, that the lower boundis closer to the truth than the upper bound. Numerical data [15, 14]up to p < 10000 suggest (very tentatively) that the correct order ofmagnitude for the clique number of Pp is c log2 p (see the discussionand the plot of the function s(p) at [16]).In this note we prove the slightly improved upper bound s(p) ≤√p−1 for the majority of the primes p = 4k+1 (we will often suppressthe dependence on p, and just write s instead of s(p)).We will denote the set of nonzero quadratic residues by Q, and thatof nonzero non-residues by NQ. Note that 0 /∈ Q and 0 /∈ NQ.2. The improved upper boundTheorem 2.1. Let q be a prime-power, q = pk, and assume that k isodd and q ≡ 1 (mod 4). Let s = s(q) be the maximal cardinality of aset B ⊂ Fq such that the difference set B − B contains only squares.(i) If [√q] is even then s2 + s− 1 ≤ q,(ii) if [√q] is odd then s2 + 2s− 2 ≤ q.Proof. The claims hold if s < [√q]. Hence we may assume that s ≥[√q].Lemma 2.2. Let D ⊂ Fq be a set such thatD ⊂ NQ, D −D ⊂ Q ∪ {0}.With r = |D| we have(1) s(q) ≤ 1 + q − 12r.SQUARES AND DIFFERENCE SETS IN FINITE FIELDS 3Proof. Let B be a maximal set such that B − B ⊂ Q ∪ {0}, |B| =s(q) = s. Consider the equationb1 − b2 = zd, b1, b2 ∈ B, d ∈ D, z ∈ NQ.This equation has exactly s(s − 1)r solutions; indeed, every pair ofdistinct b1, b2 ∈ B and a d ∈ D determines z uniquely. On the otherhand, given b1 and z, there can be at most one pair b2 and d to form asolution. Indeed, if there were another pair b′2, d′, then by substractingthe equationsb1 − b2 = zd, b1 − b′2 = zd′we get (b′2− b2) = z(d − d′), a contradiction, as the left hand side is asquare and the right hand side is not. This gives s(s−1)r ≤ s(q−1)/2as wanted. We try to construct such a set D in the form D = (B− t)∩NQ witha suitable t. The required property then follows from D−D ⊂ B−B.Let χ denote the quadratic multiplicative character, i.e. χ(t) = ±1according to whether t ∈ Q or t ∈ NQ (and χ(0) = 0). Let(2) ϕ(t) =∑b∈Bχ(b− t).Clearlyϕ(t) = |(B − t) ∩Q| − |(B − t) ∩NQ|,and hence for t /∈ B we have|(B − t) ∩NQ| = s− ϕ(t)2.To find a large set in this form we need to find a negative value of ϕ.We list some properties of this function. For t ∈ B we have ϕ(t) =s− 1, and otherwiseϕ(t) ≤ s− 2, ϕ(t) ≡ s (mod 2)(the inequality expresses the maximality of B). Furthermore,∑tϕ(t) = 0,and, since translations of the quadratic character have the quasi-orthogonalityproperty ∑tχ(t+ a)χ(t + b) = −1for a 6= b, we conclude∑tϕ(t)2 = s(q − 1)− s(s− 1) = s(q − s).4 C. BACHOC, M. MATOLCSI, AND I. Z. RUZSABy substracting the contribution of t ∈ B we obtain∑t/∈Bϕ(t) = −s(s− 1),∑t/∈Bϕ(t)2 = s(q − s)− s(s− 1)2 = s(q − s2 + s− 1).These formulas assume an even nicer form by introducing the func-tion ϕ1(t) = ϕ(t) + 1:(3)∑t/∈Bϕ1(t) = q − s2,(4)∑t/∈Bϕ1(t)2 = (s+ 1)(q − s2).As a byproduct, the second equation shows the familiar estimate s ≤√q, so we have s = [√q] <√q (recall that we assume that s ≥ [√q],the theorem being trivial otherwise).Now we consider separately the cases of odd and even s. If s is even,then, since∑t/∈B ϕ(t) < 0 and each summand is even, we can find at with ϕ(t) ≤ −2. This gives us an r with r ≥ (s + 2)/2, and onsubstituting this into (1) we obtain the first case of the theorem.If s is odd, we claim that there is a t with ϕ(t) ≤ −3. Otherwisewe have ϕ(t) ≥ −1, that is, ϕ1(t) ≥ 0 for all t /∈ B. We also knowϕ(t) ≤ s− 2, ϕ1(t) ≤ s− 1 for t /∈ B. Consequently∑t/∈Bϕ1(t)2 ≤ (s− 1)∑t/∈Bϕ1(t) = (s− 1)(q − s2),a contradiction to (4). (Observe that to reach a contradiction we needthat q− s2 is strictly positive. In case of an even k it can happen thatq = s2 and the function ϕ1 vanishes outside B.)This t provides us with a setD with r ≥ (s+3)/2, and on substitutingthis into (1) we obtain the second case of the theorem. Remark 2.3. An alternative proof for the case q = p and s beingodd is as follows. Assume by contradiction that ϕ1 is even-valued andnonnegative. Then by (3) it must be 0 for at leastq − |B| − q − s22=q + s2 − 2s2values of t. Let χ˜, ϕ˜, ϕ˜1 denote the images of χ, ϕ, ϕ1 in Fq (i.e. thefunctions are evaluated mod p). By the previous observation ϕ˜1 hasat least (q+s2−2s)/2 zeroes. On the other hand, we have χ˜(x) = x q−12 ,and hence ϕ˜1 is a polynomial of degree (q−1)/2; its leading coefficientSQUARES AND DIFFERENCE SETS IN FINITE FIELDS 5is s = [√q] 6= 0 mod p (This last fact may fail if q = pk, even if k isodd. Therefore this proof is restricted in its generality. Nevertheless weinclude it here, because we believe that it has the potential to lead tostronger results if q = p.) Consequently ϕ˜1 can have at most (q− 1)/2zeros, a contradiction. In the case of even k we can have s =√q ≡ 0(mod p) and so the polynomial ϕ˜1 can vanish, as it indeed does whenB is a subfield.Remark 2.4. It is clear from (1) that any improved lower bound on rwill lead to an improved upper bound on s. If one thinks of elements ofZp as being quadratic residues randomly with probability 1/2, then weexpect that r ≥ s2+c√s. This would lead to an estimate s ≤ √p−cp1/4.This seems to be the limit of this method. In order to get an improvedlower bound on r one can try to prove non-trivial upper bounds onthe third moment∑t∈Zpϕ3(t). To do this, we would need that thedistribution of numbers b1−b2b1−b3is approximately uniform on Q as b1, b2, b3ranges over B. This is plausible because if s ≈ √p then the distributionof B−B must be close to uniform on NQ. However, we could not proveanything rigorous in this direction.Remark 2.5. Theorem 2.1 gives the bound s ≤ [√p]−1 for about threequarters of the primes p = 4k + 1. Indeed, part (ii) gives this boundfor almost all p such that n = [√p] is odd, with the only exceptionwhen p = (n+ 1)2 − 3. Part (i) gives the improved bound s ≤ n− 1 ifn2 + n− 1 > p. This happens for about half of the primes p such thatn is even.AcknowledgmentThe authors are grateful to Pe´ter Csikva´ri for insightful commentsregarding the prime-power case.References[1] A. Blokhuis, On subsets of GF (q2) with square differences, IndagationesMathematicae, vol. 87, no. 4, pp. 369-372, (1984).[2] B. Bolloba´s, Random Graphs, (second ed.), Cambridge University Press,Cambridge, 2001.[3] P. J. Cameron, Automorphism groups in graphs, in: R. J. Wilson, L. W.Beineke (Eds.), Selected Topics in Graph Theory, vol. 2, Academic Press,NewYork, (1983), pp. 89-127.[4] F. R. K. Chung, R. L. Graham, R. M. Wilson, Quasi-random graphs,Combinatorica, Volume 9, Issue 4, (1989), pp 345-362.[5] S. D. Cohen, Clique numbers of Paley graphs, Quaestiones Math. 11, (2)(1988), 225-231.6 C. BACHOC, M. MATOLCSI, AND I. Z. RUZSA[6] E. Croot, V. Lev, Open problems in additive combinatorics, Additive com-binatorics CRM Proc. Lecture Notes Amer. Math. Soc., Providence, RI, 43,(2007), 207-233.[7] P. Delsarte, An algebraic approach to the association schemes of codingtheory, Philips Res. Rep. Suppl. 10 (1973).[8] S. Graham, C. Ringrose, Lower bounds for least quadratic non-residues,Analytic Number Theory (Allterton Park, IL, 1989), 269-309.[9] M. Krivelevich, B. Sudakov, Pseudo-random graphs, in: More Sets,Graphs and Numbers, Bolyai Society Mathematical Studies 15, Springer,(2006), 199-262.[10] J. H. van Lint, R. M. Wilson, A Course in Combinatorics, CambridgeUniversity Press, Cambridge, 1992 (2nd edition in 2001).[11] L. Lova´sz, Combinatorial Problems and Exercises, North-Holland, Amster-dam, 1979 (2nd edition in 1993).[12] E. Maistrelli, D. B. Penman, Some colouring problems for Paley graphs,Discrete Mathematics 306 (2006) 99-106.[13] M. Matolcsi, I. Z. Ruzsa, Difference sets and positive exponen-tial sums I. General properties, preprint submitted for publication,http://arxiv.org/abs/1207.1781[14] Web-page of Geoffrey Exoo with clique numbers of Paley graphs for 7000 <p < 10000, http://ginger.indstate.edu/ge/PALEY/[15] Web-page of J. B. Shearer with clique numbers of Paley graphs for p < 7000,http://www.research.ibm.com/people/s/shearer/indpal.html[16] Discussion of clique numbers and plot of the function s(p) for p < 10000,http://mathoverflow.net/questions/48591/cliques-paley-graphs-and-quadratic-residuesC. B.: Univ Bordeaux, Institut de Mathe´matiques de Bordeaux, 351,cours de la Libe´ration 33405, Talence cedex, France, Tel: (+33) 0540 00 21 61, Fax: (+33) 05 40 00 21 23E-mail address : bachoc@math.u-bordeaux1.frM. M.: Alfe´d Re´nyi Institute of Mathematics, Hungarian Academyof Sciences POB 127 H-1364 Budapest, Hungary Tel: (+361) 483-8307,Fax: (+361) 483-8333E-mail address : matomate@renyi.huI. Z. R.: Alfre´d Re´nyi Institute of Mathematics, Hungarian Acad-emy of Sciences POB 127 H-1364 Budapest, Hungary Tel: (+361) 483-8328, Fax: (+361) 483-8333E-mail address : ruzsa@renyi.hu

A Course in Combinatorics,

An algebraic approach to the association schemes of coding theory,

Automorphism groups in graphs, in:

Clique numbers of Paley graphs,

Combinatorial Problems and Exercises, North-Holland,

Diﬀerence sets and positive exponential sums I. General properties, preprint submitted for publication,

Lower bounds for least quadratic non-residues, Analytic Number Theory (Allterton Park,

Squares and difference sets in finite fields

Abstract

Similar works

Full text

Available Versions

Repository of the Academy's Library