DSpace at VNU: Monochromatic sum and product in Z mZ

Accepted Manuscript Monochromatic sum and product in Z/mZ Le Anh Vinh PII: DOI: Reference: S0022-314X(14)00090-0 10.1016/j.jnt.2014.02.004 YJNTH 4815 To appear in: Journal of Number Theory Received date: 19 September 2012 Revised date: February 2014 Accepted date: 19 February 2014 Please cite this article in press as: L Anh Vinh, Monochromatic sum and product in Z/mZ, J Number Theory (2014), http://dx.doi.org/10.1016/j.jnt.2014.02.004 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain Monochromatic sum and product in Z/mZ Le Anh Vinh∗ University of Education Vietnam National University, Hanoi vinhla@vnu.edu.vn Abstract Shkredov (2010) showed that if the finite field Zp , where p is a prime, is colored in an arbitrary way in finitely many colors, then there are x, y ∈ Zp such that x + y, xy have the same color Cilleruelo (2011) extended this result to arbitrary finite fields using Sidon sets In this short note, we present a graph-theoretic proof of this result Using the same techniques, we extend this result in the setting of the finite cyclic ring Introduction Let k be a positive integer, and χ be an arbitrary coloring of positive integers with k colors More formally, let χ : Z → {1, 2, , k} be an arbitrary map and we associate the segment of positive integers {1, 2, , k} with k different colors If f (x1 , , xn ) = 0, xi ∈ Z is an (0) (0) (0) equation, then a solution (x1 , , xn ) of the equation is called monochromatic if all xi (0) have the same color In other words, there exists m ∈ {1, , k} such that χ(xi ) = m, i = 1, , n The problem of finding monochromatic solutions of some equations was considered extensively in the literature, see for example, [1], [5]–[15] A classical result giving a complete answer about monochromatic solutions of any linear equation is Rado’s theorem (see [12]) Monochromatic solutions of linear equations and system of linear equations were studied not only in Z but in another groups (see e.g [7, 1, 5, 14] and the references therein) On the other hand, we know a little about monochromatic solutions of nonlinear equations For example, there is no answer yet to the question about monochromatic solutions of the equation x2 + y = z nor of the system x + y, xy having the same color for any finite coloring of Z (see [9, Problem 3]) Shkredov [15] used Weil’s bound for exponential sums with multiplicative characters to give a positively answer to the later question in the case of the prime field More precisely, Shkredov showed that if p is a prime number and A1 , A2 ⊂ Zp be any sets, |A1 ||A2 | ≥ 20p, then there exist x, y ∈ Zp such that x + y ∈ A1 , xy ∈ A2 Cilleruelo [4] extended this ∗ This research was supported by Vietnam National University - Hanoi project QGTD.13.02 result to arbitrary finite fields using Sidon sets Let q be any odd prime power and Fq be the finite field of q elements Cilleruelo showed that if X1 , X2 ⊂ Fq , |X1 ||X2 | > 2q, then there exist x, y ∈ Fq such that x + y ∈ X1 and xy ∈ X2 In this short note, we will present a graph-theoretic proof of this result under the slightly weaker condition |X1 ||X2 | > 8q Theorem 1.1 Let q be an odd prime power and Fq be the finite field of q elements For any X1 , X2 ⊂ Fq of cardinality |X1 ||X2 | > 8q, there exist x, y ∈ Fq such that x + y ∈ X1 and xy ∈ X2 Using the same techniques, we extend this result in the setting of the finite cyclic ring Let m be a large integer and Zm = Z/mZ be the ring of residues mod m Let γ(m) be the smallest prime divisor of m, τ (m) be the number of divisors of m, and φ(m) be the Euler’s totient function We identify Zm with {0, 1, , m − 1} Define the set of units and the set of nonunits in Zm by Z× m and Zm respectively We have the following finite ring analogue of Theorem 1.1 Theorem 1.2 Let m be an odd integer and Zm be the cyclic ring of m elements For any X , X2 ⊂ Z × m of cardinality 2m4 (τ (m))4 , |X1 ||X2 | > (φ(m))2 γ(m) there exist x, y ∈ Z× m such that x + y ∈ X1 and xy ∈ X2 1/2 √ Suppose that Zm is colored by less than φ(m)γ(m) colors Let X1 ≡ X2 be the largest 2m(τ (m))2 monochromatic subset of Zm then X1 , X2 satisfy the condition of Theorem 1.2 This implies that there exist x, y ∈ Z× m such that x + y, xy have the same color Note that this 1/2 √ result is more effective when γ(m) m1/ for some > as φ(m)γ(m) 2m(τ (m))2 Sum-product graphs For a graph G of order n, 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 Since G is a d-regular graph, d is an eigenvalue of its adjacency matrix with the all-one eigenvector If the graph G is connected, the eigenvalue d has multiplicity one Furthermore, if G is not bipartite, for any other eigenvalue θ of G, we have |θ| < d Let v θ denote the corresponding eigenvector of θ We will make use of the trick that v θ ∈ 1⊥ , so Jv θ = where J is the all-one matrix of size n × n (see [3] for more background on spectral graph theory) It is well known (see [2, 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 recall the following well-known fact (see, for example, [2]) Lemma 2.1 ([2, Corollary 9.2.5]) Let G = (V, E) be an (n, d, λ)-graph For any two sets B, C ⊂ V , we have d|B||C| e(B, C) − ≤ λ |B||C| n 2.1 Sum-product graphs over finite fields For any λ ∈ Fq , the sum-product graph SP q (λ) is defined as follows The vertex set of the sum-product graph SP q (λ) is the set Fq × Fq Two vertices U = (a, b) and V = (c, d) ∈ V (SP q (λ)) are connected by an edge, (U, V ) ∈ E(SP q (λ)), if and only if a + c + λ = bd Note that our construction is similar to that of Solymosi in [16] We have the following pseudo-randomness of the sum-product graph SP q (λ) Theorem 2.2 The graph SP q (λ) is an q , q, q 1/2 − graph Proof It is clear that SP q (λ) is a regular graph of order q and of valency q We now estimate the eigenvalues of this multigraph (we allow the multigraph to have loops but not to have multiple edges between the same two vertices) For any U = (a, b) = V = (c, d) ∈ V (SP q (λ)), we count the number of solutions of the following system a + u + λ = bv, c + u + λ = dv, (u, v) ∈ V (SP q (λ)) The system has a unique solution ad − bc − λ, b−d a−c v = b−d u = if b = d, and no solution otherwise In other words, two different vertices U = (a, b) and V = (c, d) have a unique common vertex if b = d and no common vertex otherwise Let A be the adjacency matrix of SP q (λ) For any two vertices U, V then (A2 )U,V is the number of common vertices of U and V It follows that A2 = J + (q − 1)I − E, where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of thegraph SE , where V (SE ) = Fq × Fq and for any two difference vertices a, b ∈ V (SE ), (a, b) is an edge of SE if and only if a2 = b2 It follows that SE is a q-regular graph Since SP q (λ) is a q-regular graph, q is an eigenvalue of A with the all-one eigenvector The graph SP q (λ) is connected, therefore the eigenvalue q has multiplicity one It is clear that SP q contains (many) triangles which implies that the graph is not bipartite Hence, for any other eigenvalue θ of SP q , |θ| < q Let v θ denote the corresponding eigenvector of θ Note that v θ ∈ 1⊥ , so Jv θ = Therefore, we have (θ2 − q + 1)v θ = −Ev θ , (2.1) and v θ is also an eigenvector of E By the definition of E, the graph SE is a disjoint union of q copies of the complete graph Kq This implies that SE has eigenvalues q − with multiplicity q, and −1 with multiplictity q(q − 1) One corresponding eigenvector of q − is the all-one eigenvector and all other corresponding eigenvectors can be chosen in the orthogonal space 1⊥ Plug in to Eq (2.1), A has eigenvalues q with multiplicity √ √ 1, with multiplicity q − 1, and others eigenvalues are q or − q Besides, SP q has q loops so sum of eigenvalues are equal to the trace q of A It follows that the multiplicity √ √ of q is equal to the multiplicity of − q, concluding the proof of the theorem 2.2 Sum-product graphs over finite rings For any λ ∈ Zm , the sum-product graph SP m (λ) is defined as follows The vertex set of the sum-product graph SPm (λ) is the set V (SPm (λ)) = Zm × Zm Two vertices U = (a, b) and V = (c, d) ∈ V (SP m (λ)) are connected by an edge, (U, V ) ∈ E(SP m (λ)), if and only if a + c + λ = bd We have a finite ring analogue of Theorem 2.2 Theorem 2.3 For any λ ∈ Zm , the sum-product graph, SP m (λ), is a m2 , m, 2τ (m) m γ(m)1/2 − graph Proof It is easy to see that SP m (λ) is a regular graph of order m2 and valency m We now compute the eigenvalues of this multigraph For any a, b, c, d ∈ Zm , we count the number of solutions of the following system a + u + λ = bv, c + u + λ = dv, u, v ∈ Zm (2.2) It is clear that the equation has no solution when b = d For any b = d ∈ Zm , for each solution v of (b − d)v = a − c, (2.3) there exists a unique u satisfying the system (2.2) Therefore, we only need to count the number of solutions of (2.3) Let n be the largest divisor of m such that b − d is also divisible by n over the ring Zm If n a − c then Eq (2.3) has no solution Suppose that × n | a − c Let μ = (a − c)/n ∈ Zm/n and x = (b − d)/n ∈ Z× m/n Since x ∈ Zm/n , there exists unique v ∗ ∈ Zm/n such that xv ∗ = μ mod m/n For each v ∗ , putting back into Eq (2.3) gives us n solutions Hence, Eq (2.3) has n solutions if n | a − c Therefore, for any two vertices U = (a, b) and V = (c, d) ∈ V (SP m (λ)), let n be the largest divisor of m such that b − d is also divisible by n, then U and V have n common neighbors if n | c − a and no common neighbors otherwise Let A be the adjacency matrix of SP m (λ) For any two vertices U, V then (A2 )U,V is the number of common vertices of U and V It follows that A2 = J + (m − 1)I − (n − 1)Fn , En + n|m 1≤n