DSpace at VNU: Monochromatic sum and product in Z mZ

Monochromatic sum and product in Z/mZLe Anh Vinh∗ University of Education Vietnam National University, Hanoi vinhla@vnu.edu.vn Abstract Shkredov 2010 showed that if the finite fieldZp, whe

Accepted Manuscript

Monochromatic sum and product inZ/mZ

Le Anh Vinh

PII: S0022-314X(14)00090-0

DOI: 10.1016/j.jnt.2014.02.004

Reference: YJNTH 4815

To appear in: Journal of Number Theory

Received date: 19 September 2012

Revised date: 9 February 2014

Accepted date: 19 February 2014

Please cite this article in press as: L Anh Vinh, Monochromatic sum and

product inZ/mZ, J Number Theory (2014),

http://dx.doi.org/10.1016/j.jnt.2014.02.004

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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 fieldZp, wherep is a prime, is colored

in an arbitrary way in finitely many colors, then there are x, y ∈ Z p 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

1 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, , x n ) = 0, x i ∈ Z is an equation, then a solution (x(0)1 , , x(0)n ) of the equation is called monochromatic if all x(0)i have the same color In other words, there exists m ∈ {1, , k} such that χ(x(0)i ) = 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+ y2 = z2 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⊂ Z pbe any sets,|A1||A2| ≥ 20p, then there exist x, y ∈ Z p 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 F q be

the finite field of q elements Cilleruelo showed that if X1, X2 ⊂ F q , |X1||X2| > 2q, then there exist x, y ∈ F q such that x + y ∈ X1and 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 F q be the finite field of q elements For any X1, X2 ⊂ F q of cardinality |X1||X2| > 8q, there exist x, y ∈ F q 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 Z m = 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 Z0

m respectively We have the following finite

ring analogue of Theorem 1.1

Theorem 1.2 Let m be an odd integer and Z m be the cyclic ring of m elements For any X1, X2 ⊂ Z ×

m of cardinality

|X1||X2| > 2m4(τ (m))4

(φ(m))2γ(m) , there exist x, y ∈ Z × m such that x + y ∈ X1 and xy ∈ X2

Suppose thatZmis colored by less than φ(m)γ(m)

1/2

√

2m(τ(m))2 colors Let X1≡ X2be the largest 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 result is more effective when γ(m)  m 1/ for some  > 0 as φ(m)γ(m) √ 1/2

2m(τ(m))2  1.

2 Sum-product graphs

For a graph G of order n, let λ1 ≥ λ2 ≥ ≥ λ n 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 1 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 = 0 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 

e(B,C) − d|B||C| n  ≤ λ|B||C|.

For any λ ∈ F q, the sum-product graphSP q (λ) is defined as follows The vertex set of the

sum-product graph SP q (λ) is the set F q × F q 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 graphSP q (λ).

Theorem 2.2 The graph SP q (λ) is an



q2, q, q 1/2

− 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 (we allow the multigraph to have loops but

(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

u = ad − bc

b − d − λ,

v = a − c

b − d

be the adjacency matrix ofSP 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 S E , where V (S E) = Fq × F q and for any two difference 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-regular graph.

SinceSP q (λ) is a q-regular graph, q is an eigenvalue of A with the all-one eigenvector 1.

The graphSP q (λ) is connected, therefore the eigenvalue q has multiplicity one It is clear

thatSP 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= 0 Therefore, we have

and v θ is also an eigenvector of E By the definition of E, the graph S E is a disjoint

union of q copies of the complete graph K q This implies that S E has eigenvalues q − 1 with multiplicity q, and −1 with multiplictity q(q − 1) One corresponding eigenvector of

q − 1 is the all-one eigenvector 1 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, 0 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. 

For any λ ∈ Z m, the sum-product graph SP m (λ) is defined as follows The vertex set of the sum-product graph SP m (λ) is the set V (SP m (λ)) = Z m ×Z m 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 ∈ Z m , 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 thatSP 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 ∈ Z m, we count the

number of solutions of the following system

a + u + λ = bv, c + u + λ = dv, u, v ∈ Z m (2.2)

m, for each

solution v of

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 Z m If n  a − c then Eq (2.3) has no solution Suppose that

n | a − c Let μ = (a − c)/n ∈ Z m/n and x = (b − d)/n ∈ Z × m/n Since x ∈ Z × m/n, there

exists unique v ∗ ∈ Z m/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|m

1≤n<m

E n+



n|m

1<n<m

where:

• J is the all-one matrix and I is the identity matrix of size m2;

• E n is the adjacency matrix of the graph B E,n , where for any two vertices U = (a, b) and V = (c, d) ∈ V (SP m (λ)), (U, V ) is an edge of B E,n if and only if n is the largest divisor of m such that b − d is divisible by n but c − d is not divisible by n;

• F n is the adjacency matrix of the graph B F,n , where for any two vertices U = (a, b) and V = (c, d) ∈ V (SP m (λ)), (U, V ) is an edge of B F,n if and only if n is the largest divisor of m such that b − d divisible by n and c − d is also divisible by n.

For any n > 1 then B E,n is a regular graph of valency less than m(m/n) and B F,n

is a regular graph of valency less than (m/n)2 Since eigenvalues of a regular graph are

bounded by its valency, all eigenvalues of E n are at most m(m/n) and all eigenvalues of

F n are at most (m/n)2 Note that E1 is a zero matrix.

SinceSP m (λ) is a m-regular graph, m is an eigenvalue of A with the all-one eigenvector

1 The graphSP m (λ) is connected therefore the eigenvalue m has multiplicity one Since

the graph SP m (λ) contains (many) triangles, it is not bipartite Hence, for any other

eigenvalue θ, |θ| < m Let v θ denote the corresponding eigenvector of θ Note that

v θ ∈ 1 ⊥ , so Jv θ = 0 It follows from (2.4) that

(θ2− m + 1)v θ=

⎛

⎜

n|m

1<n<m

E n − 

n|m

1<n<m (n − 1)F n

⎞

⎟

⎠ v θ

Hence, v θ is also an eigenvalue of



n|m

1<n<m

E n − 

n|m

1<n<m (n − 1)F n

Since absolute value of eigenvalues of sum of matrices are bounded by sum of largest absolute values of eigenvalues of summands We have

θ2 ≤ m − 1 + 

n|m

1<n<m

m(m/n) + 

n|m

1<n<m (n − 1)(m/n)2

< m + 2m2 

n|m

1<n<m

< 2τ (m)m2γ(m) −1

3 Monochromatic sum and product

We will give a unified proof of both Theorem 1.1 and Theorem 1.2 The proofs of both

Theorem 1.1 and Theorem 1.2 are based on the study of the equation (x1/2−z)(x1/2+z) =

x2 where x1∈ X1, x2 ∈ X2 and z ∈ X3 Here, X3≡ F q in the finite field case and X3 ≡ Z ∗

m

in the finite ring case

This equation is equivalent to the equation (x1/2)2− x2 = z2 Set B = A1× A3 and

C = A2 × A4, where

A1 = {(x1/2)2| x1 ∈ X1}, A2 = {−x2 | x2 ∈ X2}, A3 = {z2| z ∈ X3}, A4 = {z2| z ∈ X3}.

Now, we will proceed the finite field case and the finite ring case separately

It is clear that the equation (x1/2)2− x2 = z2 has a solution x1 ∈ X1, x2 ∈ X2, z ∈ X3

if and only if there exists an edge between two vertex sets B and C of the sum-product

graph SP q(0) It follows from Theorem 2.2 and Theorem 2.1 that the number of edges

between B and C is bounded below by

e(B, C) ≥ |A1||A2||A3||A4|

Note that|A1| ≥ |X1|/2, |A2| = |X2| and |A3| = |A4| = (q + 1)/2 This implies that

|A1||A2||A3||A4| ≥ |X1||X2|(q + 1)2/8 > q3. (3.2)

Putting (3.1) and (3.2) together, we have e(B, C) > 0, concluding the proof of

Theo-rem 1.1

The equation (x1/2)2− x2 = z2 has a solution x1 ∈ X1, x2 ∈ X2, z ∈ X3 if and only if

there exists an edge between two vertex sets B and C of the sum-product graph SP q(0).

It follows from Theorem 2.3 and Theorem 2.1 that the number of edges between B and

C is bounded below by

e(B, C) ≥ |A1||A2||A3||A4|

m −2τ (m)m2γ(m) −1 |A1||A2||A3||A4|. (3.3)

Let m = p α1

1 p α k

k be the unique prime factorization of m then τ (m) = (α1 +

1) (α k + 1) ≥ 2 k It is easy to see that a2 ≡ b2 mod m if and only if a ≡ ±b

mod p α i

i for any i = 1, , k Hence, |A1| ≥ |X1|/2k ≥ |X1|/τ(m), |A2| = |X2| and

|A3| = |A4| ≥ φ(m)/2 k ≥ φ(m)/τ(m) This implies that

|A1||A2||A3||A4| ≥ |X1||X2|(φ(m))2/τ (m)3> 2m

4τ (m)

Putting (3.3) and (3.4) together, we have e(B, C) > 0, concluding the proof of

Theo-rem 1.2

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