Journal of Combinatorial Theory, Series B 103 (2013) 651–657 Contents lists available at ScienceDirect Journal of Combinatorial Theory, Series B www.elsevier.com/locate/jctb Graphs generated by Sidon sets and algebraic equations over finite fields Le Anh Vinh University of Education, Vietnam National University, Hanoi, Viet Nam a r t i c l e i n f o Article history: Received 29 December 2011 Available online 13 September 2013 Keywords: Sidon sets Sum-product estimates Sum-product graphs a b s t r a c t We study the spectra of several graphs generated by Sidon sets and algebraic equations over finite fields These graphs are used to study some combinatorial problems in finite fields, such as sum product estimates, solvability of some equations and the distribution of their solutions © 2013 Elsevier Inc All rights reserved Introduction Let X be a finite abelian group For any sets A , B ⊂ X and x ∈ X , we write r A − B (x) for the number of representations of x = a − b, a ∈ A, b ∈ B We say that a set A ⊂ X is a Sidon set if r A − A (x) = By counting the number of differences of a − a , we can see that if A is a Sidon set, whenever x√ then | A | √ < | X | + 1/2 The most interesting Sidon sets are those which have large cardinality, that is, | A | = | X | − δ where δ is a small number Note that, throughout this paper, the cardinality | X | is viewed as an asymptotic parameter In [4], Cilleruelo introduced a new elementary method to study a class of combinatorial problems in finite fields: sum-product estimates [6,10], solvability of some equations [12,13], distribution of sequences in small intervals [5,7–9,11], incidence problems [14,15], etc More precisely, Cilleruelo proved the following theorem, which is the main tool in his method Theorem 1.1 (See [4, Theorem 2.1].) Let A be a Sidon set in a finite abelian group X with | A | = Then, for all B , B ⊂ X , we have √ | X | − δ E-mail address: vinhla@vnu.edu.vn This research was supported by Vietnam National Foundation for Science and Technology Development grant 101.01-2011.28 0095-8956/$ – see front matter © 2013 Elsevier Inc All rights reserved http://dx.doi.org/10.1016/j.jctb.2013.07.003 652 L.A Vinh / Journal of Combinatorial Theory, Series B 103 (2013) 651–657 b, b ∈ B × B , b + b ∈ A = | A| |B| B + θ |B| B |X| 1/2 | X |1/4 , |B| with |θ| < + | X | max(0, δ) In this paper, we first obtain a version of Theorem 1.1 using a graph theory approach The use of graph spectra on combinatorial problems in finite fields was first invented by Van Vu [16], then adapted by the author [15] and Solymosi [14] Our approach here was based on Solymosi’s method [14] More precisely, in Section 2, we give a spectral proof of the following theorem Theorem 1.2 Let X be a finite abelian group of odd order and A be a Sidon set in X with | A | = Then, for all B , B ⊂ X , we have b, b ∈ B × B , b + b ∈ A where |θ| √ = | A| | B | B + θ| X |1/4 | B | B |X| 1/2 √ | X | − δ , 2(1 + δ) Note that the upper bound for |θ| in Theorem 1.2 is slightly weaker than that appearing in Theorem 1.1 Using Theorem 1.1 one can similarly with Theorem 1.2, Cilleruelo recovered and improved various results in the literature (see [4] and references therein) In Section 3, we apply our spectral method to give direct proofs of some of these results We also discuss applications of these results on various combinatorial problems over finite fields Let Fq be a finite field of q elements where q is a large odd prime power For any non-empty subsets A , B of a finite field Fq , we consider the sum set A + B := {a + b: a ∈ A , b ∈ B } and the product set A · B := {a.b: a ∈ A , b ∈ B } Let A be a subset of a prime field F p := Z/ p Z for some odd prime p From the work of Bourgain, Katz and Tao [3] with subsequent refinement by Bourgain, Glibichuk and Konyagin [2] it is known that if | A | < p 1−δ for some δ > then one has the estimate | A + A | + | A · A | c δ | A |1+c for some ˝ and Szemerédi In this paper, we obtain c = c (δ) > This is a finite field analogue of a result of Erdos some related results on sum and product sets In particular, we show that if A ⊂ Fq of cardinality √ | A | > 3q3/4 then A + A + A · A, and A · A · ( A + A ) contain all elements of Fq× = Fq \{0} Note that, the statement “ A + A + A · A contains all elements of Fq× whenever A is sufficiently large” is also implicit in the work of Sárközy [13] and Cilleruelo [4] Graphs generated by Sidon sets For a graph G, let λ1 λ2 · · · λn be the eigenvalues of its adjacency matrix The quantity λ(G ) = max{λ2 , −λn } is called the second eigenvalue of G A graph G = ( V , E ) is called an (n, d, λ)-graph if it is d-regular, has n vertices, and the second eigenvalue of G is at most λ It is well known (see [1, Chapter 9] for more details) that if λ is much smaller than the degree d, then G has certain random-like properties For two (not necessarily) disjoint subsets of vertices U , W ⊂ V , let e (U , W ) be the number of ordered pairs (u , w ) such that u ∈ U , w ∈ W , and {u , w } is an edge of G We first recall the following well-known fact (see, for example, [1]) Theorem 2.1 (See [1, Corollary 9.2.5].) Let G = ( V , E ) be an (n, d, λ)-graph For any two sets B , C ⊂ V , we have e( B , C ) − d| B ||C | n λ | B ||C | L.A Vinh / Journal of Combinatorial Theory, Series B 103 (2013) 651–657 653 Let X be a finite abelian group For any Sidon set A ⊂ X , the Cayley graph SG A , X on X is defined as follows The vertex set V (SG A , X ) of the graph SG A , X is the group X Two vertices a and b ∈ V (SG A , X ) are connected by an edge, {a, b} ∈ E (SG A , X ), if and only if a + b ∈ A We have the following lemma Lemma 2.2 Let X be a finite abelian group of odd order For any Sidon set A ⊂ X with | A | = graph SG A , X is connected and non-bipartite √ | X | − δ , the Proof Let a1 , a2 , a3 be three distinct elements of the Sidon set A The graph SG A , X contains the triangle with three vertices (a1 + a2 − a3 )/2, (a2 + a3 − a1 )/2, and (a3 + a1 − a2 )/2 (by the structure theorem on finite abelian groups, note that the order of X is odd, one can divide a given element of X by 2) This implies that SG A , X is a non-bipartite graph We now prove that the graph SG A , X is connected by showing that there is a path of length four between any two vertices of the graph For a ∈ V (SG A , X ), let N (a) = b ∈ V (SG A , X ) {a, b} ∈ E (SG A , X ) be the set of neighbors of a in SG A , X For any b1 = b2 ∈ N (a), we will show that N (b1 ) ∩ N (b2 ) ≡ {a} Suppose that there exists c = a ∈ N (b1 ) ∩ N (b2 ) Note that a + b1 , a + b2 , c + b1 , c + b2 ∈ A, so we have a − c = (a + b1 ) − (c + b1 ) = (a + b2 ) − (c + b2 ) ∈ A − A This implies that r A − A (a − c ) > 1, which is a contradiction Let N (a) = N (b) b∈ N (a) be the set of vertices that can be reached from a by a path of length two Since N (b1 ) ∩ N (b2 ) ≡ {a} for any b1 = b2 ∈ N (a), we have N (a) = + | A | | A | − = | A |2 − | A | + For any a = b ∈ X , we have N (a) + N (b) = 2| A |2 − 2| A | + > | X |, which implies that N (a) ∩ N (b) ≡ ∅ Therefore, b can be reached from a by a path of length four It concludes the proof of the lemma ✷ Now, we can prove the following pseudo-randomness of the Cayley graph SG A , X Theorem 2.3 Let X be a finite abelian group For any Sidon set A ⊂ X with | A | = is an √ | X | − δ , the graph SG A , X | X |, | A |, 2(1 + δ)| X |1/2 -graph Proof Our proof of Theorem 2.3 is based on the work of Solymosi in [14] It is clear that SG A , X is a regular graph of order | X | and of valency | A | We now estimate the eigenvalues of this multigraph (i.e graph with loops) For any a = b ∈ X , we count the number of solutions of the following system a + x, b + x ∈ A , x ∈ X (2.1) Since A is a Sidon set, there exists at most one representation of a − b in the set A − A Hence, the system (2.1) has a unique solution if a − b ∈ A − A and no solution otherwise In other words, two 654 L.A Vinh / Journal of Combinatorial Theory, Series B 103 (2013) 651–657 different vertices a and b have a unique common vertex if a − b ∈ A − A and no common vertex otherwise Let M be the adjacency matrix of SG A , X It follows that M = J + | A| − I − E , (2.2) where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph S E , where V (S E ) = X and for any two distinct vertices a, b ∈ X , {a, b} is an edge of S E if and only if / A − A It follows from A is a Sidon set that S E is a (| X | − − | A |(| A | − 1))-regular graph a−b∈ Besides, SG A , X is a | A |-regular graph, | A | is an eigenvalue of SG A , X with all-one eigenvector From Lemma 2.2, the graph SG A , X is connected and non-bipartite so the eigenvalue | A | has multiplicity one; and for any other eigenvalue θ of SG A , X , |θ| < | A | Let v θ denote the corresponding eigenvector of θ Note that v θ ∈ 1⊥ , so J v θ = It follows from (2.2) that (θ − | A | + 1) v θ = − E v θ Since S E is a (| X | − − | A |(| A | − 1))-regular graph, absolute values of eigenvalues of S E are at most | X | − − | A |(| A | − 1) This implies that θ2 | A | − + | X | − − | A | | A | − < 2(1 + δ)| X |1/2 The theorem follows ✷ Theorem 1.2 is just an immediate corollary of Theorem 2.1 and Theorem 2.3 See also [14] for another graph generated on a Sidon set in Fq Graphs generated by equations over finite fields In this section, we study the pseudo-randomness of three graphs generated by equations over finite fields Note that, Theorem 3.1, Theorem 3.3, and Theorem 3.6 can be obtained as corollaries of Theorem 2.3 Indeed, all is needed is to prove that the sets involved are Sidon sets and it was done explicitly in the paper of Cilleruelo [4] 3.1 Sum-square graphs The sum-square graph SS q is defined as follows The vertex set of the product graph S S q is the set Fq × Fq Two vertices a = (a1 , a2 ) and b = (b1 , b2 ) ∈ V (SS q ) are connected by an edge, {a, b} ∈ E (SS q ), if and only if a1 + b1 = (a2 + b2 )2 We have the following pseudo-randomness of the sum-square graph SS q Theorem 3.1 The graph SS q is an q2 , q, 2q -graph Proof It is clear that SS q is a regular graph of order q2 and of valency q We now estimate the eigenvalues of this multigraph (i.e graph with loops) For any a = (a1 , a2 ) = b = (b1 , b2 ) ∈ V (SS q ), we count the number of solutions of the following system a1 + x1 = (a2 + x2 )2 , b1 + x1 = (b2 + x2 )2 , x = (x1 , x2 ) ∈ V (SS q ) The system has a unique solution x1 = x2 = a1 − b a2 − b a1 − b a2 − b 2 + (a2 − b2 ) /4 − a , − (a2 + b2 ) /2 if a2 = b2 , and no solution otherwise In other words, two different vertices a = (a1 , a2 ) and b = (b1 , b2 ) have a unique common vertex if a2 = b2 and no common vertex otherwise Let M be the adjacency matrix of SS q It follows that L.A Vinh / Journal of Combinatorial Theory, Series B 103 (2013) 651–657 M = J + (q − 1) I − E , 655 (3.1) where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph S E , where V (S E ) = Fq × Fq and for any two distinct vertices a, b ∈ V (S E ), {a, b} is an edge of S E if and only if a2 = b2 It follows that S E is a (q − 1)-regular graph Since SS q is a q-regular graph, q is an eigenvalue of M with the all-one eigenvector The graph SS q is connected, therefore the eigenvalue q has multiplicity one It is clear that SS q contains (many) triangles which implies that the graph is not bipartite Hence, for any other eigenvalue θ of SS q , |θ| < q Let v θ denote the corresponding eigenvector of θ Note that v θ ∈ 1⊥ , so J v θ = It follows from (3.1) that (θ − q + 1) v θ = − E v θ Since S E is a (q − 1)-regular graph, absolute values of eigenvalues of S E are bounded by q − This implies that θ 2(q − 1) The theorem follows ✷ We have an immediate consequence of Theorem 2.1 and Theorem 3.1 Note that, our result also follows from [4, Theorem 4.1] by taking X (x) = {x: (x, x ) ∈ U } and Y ( y ) = { y: ( y , y ) ∈ V } Corollary 3.2 Let U , V ⊂ Fq × Fq Then, the number of solutions of the equation x1 + x2 = (x3 + x4 )2 , (x1 , x3 ) ∈ U , (x2 , x4 ) ∈ V is S= |U || V | q + θ q|U || V |, where θ < Set U = (− A ) × A and V = (− A + λ) × A for a large subset A of the field Fq , we conclude that the set B + B · B contains the whole field when B = A + A (see [8] for the expanding property of A ( A + 1) for an arbitrary subset A ⊂ Fq ) 3.2 Sum-product graphs For any λ ∈ Fq , the sum-product graph SP q (λ) is defined as follows The vertex set of the sumproduct graph SP q (λ) is the set Fq × Fq Two vertices a = (a1 , a2 ) and b = (b1 , b2 ) ∈ V (SP q (λ)) are connected by an edge, {a, b} ∈ E (SP q (λ)), if and only if a1 + b1 + a2 b2 = λ Note that our construction is similar to that of Solymosi in [14] We have the following pseudo-randomness of the sum-product graph SP q (λ) Theorem 3.3 The graph SP q (λ) is an q2 , q, 2q -graph Proof It is clear that SP q (λ) is a regular graph of order q2 and of valency q We now estimate the eigenvalues of this multigraph (i.e graph with loops) For any a = (a1 , a2 ) = b = (b1 , b2 ) ∈ V (SP q (λ)), we count the number of solutions of the following system a1 + x1 + a2 x2 = b1 + x1 + b2 x2 = λ, The system has a unique solution x1 = λ − x2 = a2 b − a1 b a2 − b b − a1 a2 − b , x = (x1 , x2 ) ∈ V SP q (λ) 656 L.A Vinh / Journal of Combinatorial Theory, Series B 103 (2013) 651–657 if a2 = b2 , and no solution otherwise In other words, two different vertices a = (a1 , a2 ) and b = (b1 , b2 ) have a unique common vertex if a2 = b2 and no common vertex otherwise Let M be the adjacency matrix of SP q (λ) It follows that M = J + (q − 1) I − E , where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph S E , where V (S E ) = Fq × Fq and for any two distinct vertices a, b ∈ V (S E ), {a, b} is an edge of S E if and only if a2 = b2 It follows that S E is a (q − 1)-regular graph Since SP q (λ) is a q-regular graph, q is an eigenvalue of M with the all-one eigenvector The graph SP q (λ) is connected, therefore the eigenvalue q has multiplicity one Similar to the proof of Theorem 3.1, for any other eigenvalue θ of SP q , θ < 2q − The theorem follows ✷ We have an immediate corollary of Theorem 2.1 and Theorem 3.3 Note that, our result also follows from [4, Theorem 4.1] Corollary 3.4 Let U , V ⊂ Fq × Fq For any λ ∈ Fq , the number of solutions of the equation x1 + x2 + x3 x4 = λ, (x1 , x3 ) ∈ U , (x2 , x4 ) ∈ V is S= |U || V | q + θ q|U || V |, where θ < Taking U = V = A × A for a large subset A of the field Fq , we conclude that the sum of the sum set A + A and the product set A · A contains the whole field Corollary 3.5 For any A ⊂ Fq of cardinality | A | > √ 2q3/4 , then A + A + A · A ≡ Fq 3.3 Product-sum graphs For any λ ∈ Fq× , the product-sum graph PS q (λ) is defined as follows The vertex set of the product graph PS q (λ) is the set Fq× × Fq Two vertices a = (a1 , a2 ) and b = (b1 , b2 ) ∈ V (PS q (λ)) are connected by an edge, {a, b} ∈ E (PS q (λ)), if and only if a1 b1 (a2 + b2 ) = λ We have the following pseudo-randomness of the product graph PS q (λ) Theorem 3.6 The graph PS q (λ) is an (q − 1)q, q − 1, 3q -graph Proof It is clear that PS q (λ) is a regular graph of order (q − 1)q and valency q − We now estimate the eigenvalues of this multigraph (i.e graph with loops) For any a = (a1 , a2 ) = b = (b1 , b2 ) ∈ V (PS q (λ)), we count the number of solutions of the following system a1 x1 (a2 + x2 ) = b1 x1 (b2 + x2 ) = λ, The system has a unique solution λ(b1 − a1 ) , (a2 − b2 )a1 b1 a1 a2 − b b x2 = b − a1 x1 = x = (x1 , x2 ) ∈ V PS q (λ) L.A Vinh / Journal of Combinatorial Theory, Series B 103 (2013) 651–657 657 if a1 = b1 and a2 = b2 , and no solution otherwise In other words, two different vertices a = (a1 , a2 ) and b = (b1 , b2 ) have a unique common vertex if a1 = b1 , a2 = b2 , and no common vertex otherwise Let M be the adjacency matrix of PS q (λ) It follows that M = J + (q − 2) I − E , (3.2) where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph S E , where V (S E ) = Fq× × Fq and for any two distinct vertices a, b ∈ V (S E ), {a, b} is an edge of S E if and only if a1 = b1 or a2 = b2 It follows that S E is a (2q − 3)-regular graph Since PS q (λ) is a (q − 1)-regular graph, (q − 1) is an eigenvalue of M with the all-one eigenvector The graph PS q (λ) is connected, therefore the eigenvalue q − has multiplicity one It is clear that PS q (λ) contains (many) triangles which implies that the graph is not bipartite Hence, for any other eigenvalue θ of PS q (λ), |θ| < q − Let v θ denote the corresponding eigenvector of θ Note that v θ ∈ 1⊥ , so J v θ = It follows from (3.2) that (θ − q + 2) v θ = − E v θ Since S E is a 2(q − 1)-regular graph, absolute values of all eigenvalues of S E are at most 2(q − 1) This implies that θ 2q − + q − < 3q The theorem follows ✷ We have an immediate corollary of Theorem 2.1 and Theorem 3.6 Corollary 3.7 Let U , V ⊂ Fq× × Fq For any λ ∈ Fq× , the number of solutions of the equation x1 x2 (x3 + x4 ) = λ, (x1 , x3 ) ∈ U , (x2 , x4 ) ∈ V is S= |U || V | q + θ q|U || V |, where θ < Taking U = V = A × A for a 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Theorem 1.2 is just an immediate corollary of Theorem 2.1 and Theorem 2.3 See also [14] for another graph generated on a Sidon set in Fq Graphs generated by equations over finite fields In this section,... fields In this section, we study the pseudo-randomness of three graphs generated by equations over finite fields Note that, Theorem 3.1, Theorem 3.3, and Theorem 3.6 can be obtained as corollaries... Note that, the statement “ A + A + A · A contains all elements of Fq× whenever A is sufficiently large” is also implicit in the work of Sárközy [13] and Cilleruelo [4] Graphs generated by Sidon sets