DSpace at VNU: Graphs generated by Sidon sets and algebraic equations over finite fields

Graphs generated by Sidon sets For a graph G, let λ1 λ2 · · · λn be the eigenvalues of its adjacency matrix.. A graph G= V,E is called an n,d, λ-graph if it is d-regular, has n vertices

Contents lists available atScienceDirect Journal of Combinatorial Theory,

Series B www.elsevier.com/locate/jctb

Graphs generated by Sidon sets and algebraic

equations over finite fields

Le Anh Vinh1

University of Education, Vietnam National University, Hanoi, Viet Nam

a r t i c l e i n f o a b s t r a c t

Article history:

Received 29 December 2011

Available online 13 September 2013

Keywords:

Sidon sets

Sum-product estimates

Sum-product graphs

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

1 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) 1

whenever x=0 By counting the number of differences of a−a, we can see that if A is a Sidon set,

then |A| < √ |X| +1/2 The most interesting Sidon sets are those which have large cardinality, that

is,|A| = √ |X| − δwhereδis 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| = √ |X| − δ Then, for all B,B⊂X , we have

E-mail address:vinhla@vnu.edu.vn.

1 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.

b, b

∈B×B,b+b∈A = |A|

|X| |B| B + θ 

|B| B1/2

|X|1/4, with|θ| <1+|B|

|X|max(0, δ).

In this paper, we first obtain a version of Theorem 1.1using 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 Section2, 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| = √ |X| − δ Then, for all B,B⊂X , we have

b, b

∈B×B,b+b∈A = |A|

|X| |B| B + θ |X|1/4

|B| B1/2

, where|θ|  √2(1+ δ).

Note that the upper bound for |θ|inTheorem 1.2is slightly weaker than that appearing in Theo-rem 1.1 UsingTheorem 1.1one can do similarly withTheorem 1.2, Cilleruelo recovered and improved various results in the literature (see [4]and references therein) In Section3, 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 fieldFq, 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 Fp:= Z/pZfor 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| <p1−δ for some δ >0 then one has the estimate |A+A| + |A·A| c δ|A|1+c for some

c=c(δ) >0 This is a finite field analogue of a result of Erd ˝os and Szemerédi In this paper, we obtain

some related results on sum and product sets In particular, we show that if A⊂ Fq of cardinality

|A| > √3q34 then A+A+A·A, and A·A· (A+A)contain all elements ofF×

q = Fq\{0} Note that,

the statement “ A+A+A·A contains all elements of F×

q whenever A is sufficiently large” is also

implicit in the work of Sárközy[13]and Cilleruelo[4]

2 Graphs generated by Sidon sets

For a graph G, let λ1 λ2 · · ·  λn be the eigenvalues of its adjacency matrix The quan-tity 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 ifλ 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|.

Let X be a finite abelian group For any Sidon set A⊂X , the Cayley graphSGA , X on X is defined

as follows The vertex set V(SGA , X) of the graph SGA , X is the group X Two vertices a and b∈

V(SGA , X)are connected by an edge,{a,b} ∈E(SGA , 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| = √ |X| − δ, the graphSGA , X is connected and non-bipartite.

Proof Let a1,a2,a3 be three distinct elements of the Sidon set A The graph SGA , 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 thatSGA , X is a non-bipartite graph

We now prove that the graphSGA , X is connected by showing that there is a path of length four

between any two vertices of the graph For a∈V(SGA , X), let

N(a)= b∈V( SGA , X)  {a,b} ∈E(SGA , X) 

be the set of neighbors of a inSGA , 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 rA−A(a−c) >1, which is a contradiction

Let

N2(a)= 

b∈N ( a )

N(b)

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

N2(a) =1+ |A|  |A| −1

= |A|2− |A| +1. For any a=b∈X , we have

N2(a) +N2(b) =2|A|2−2|A| +2> |X|,

which implies that N2(a) ∩N2(b) ≡ ∅ Therefore, b can be reached from a by a path of length four It

concludes the proof of the lemma 2

Now, we can prove the following pseudo-randomness of the Cayley graphSGA , X

Theorem 2.3 Let X be a finite abelian group For any Sidon set A⊂X with|A| = √ |X| − δ, the graphSGA , X

is an



|X|, |A|, 2(1+ δ)|X|1/2

-graph.

Proof Our proof ofTheorem 2.3is based on the work of Solymosi in[14] It is clear that SGA , 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

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

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 ofSGA , X It follows that

M2= J+  |A| −1

where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graphSE,

where V(SE) =X and for any two distinct vertices a,b∈X , {a,b}is an edge of SE if and only if

a−b∈ /A−A It follows from A is a Sidon set that SE is a( |X| −1− |A|(|A| −1))-regular graph Besides,SGA , X is a|A|-regular graph,|A|is an eigenvalue ofSGA , X with all-one eigenvector 1 From

Lemma 2.2, the graph SGA , X is connected and non-bipartite so the eigenvalue |A| has multiplicity one; and for any other eigenvalueθ ofSGA , X,|θ| < |A| Let v θ denote the corresponding eigenvector

ofθ Note that v θ∈1⊥, so J v

θ =0 It follows from(2.2) that(θ2− |A| +1)v θ= −E v θ SinceSE is

a ( |X| −1− |A|(|A| −1))-regular graph, absolute values of eigenvalues ofSE are at most |X| −1−

|A|(|A| −1) This implies that

θ2 |A| −1+ |X| −1− |A|  |A| −1

<2(1+ δ)|X|1/2.

The theorem follows 2

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 inFq.

3 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 3.6can 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 SSq is defined as follows The vertex set of the product graph S Sq is the set Fq× Fq Two vertices a= (a1,a2) and b= (b1,b2) ∈V(SSq) are connected by an edge,

{a,b} ∈E(SSq), if and only if a1+b1= (a2+b2)2 We have the following pseudo-randomness of the sum-square graphSSq.

Theorem 3.1 The graphSSq is an



q2,q,

2q

-graph.

Proof It is clear that SSq 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(SSq),

we count the number of solutions of the following system

a1+x1= (a2+x2)2, b1+x1= (b2+x2)2, x= (x1,x2)∈V( SSq).

The system has a unique solution

x1= a1−b1

a2−b2+ (a2−b2)

2

/ −a1,

x2= a1−b1

a2−b2− (a2+b2)

/

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 ofSS It follows that

M2= J+ (q−1)I −E, (3.1)

where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graphSE,

where V(SE) = Fq× Fq and for any two distinct vertices a,b∈V(SE),{a,b}is an edge of SE if and

only if a2=b2 It follows thatSE is a(q−1)-regular graph SinceSSq is a q-regular graph, q is an

eigenvalue of M with the all-one eigenvector 1 The graphSSqis connected, therefore the eigenvalue

q has multiplicity one It is clear that SSq contains (many) triangles which implies that the graph

is not bipartite Hence, for any other eigenvalueθ ofSSq,|θ| <q Let v θ denote the corresponding eigenvector of θ Note that v θ ∈1⊥, so J v

θ =0 It follows from(3.1)that (θ2−q+1)v θ = −E v θ SinceSE is a(q−1)-regular graph, absolute values of eigenvalues ofSE are bounded by q−1 This implies thatθ22(q−1) The theorem follows 2

We have an immediate consequence ofTheorem 2.1andTheorem 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|,

whereθ2<2.

Set U= (−A) ×A and V= (−A+ λ) ×A for a large subset A of the fieldFq, 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 anyλ ∈ Fq, the sum-product graphSPq(λ) is defined as follows The vertex set of the sum-product graphSPq(λ)is the setFq× Fq Two vertices a= (a1,a2)and b= (b1,b2) ∈V(SPq(λ))are connected by an edge,{a,b} ∈E(SPq(λ)), if and only if a1+b1+a2b2= λ Note that our construction

is similar to that of Solymosi in[14] We have the following pseudo-randomness of the sum-product graphSPq(λ)

Theorem 3.3 The graphSPq(λ)is an



q2,q, 

2q

-graph.

Proof It is clear thatSPq(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(SPq(λ)),

we count the number of solutions of the following system

a1+x1+a2x2=b1+x1+b2x2= λ, x= (x1, x2)∈V

SPq(λ) 

.

The system has a unique solution

x1= λ −a2b1−a1b2

a2−b2 , x2=b1−a1

a2−b2

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 ofSPq(λ) It follows that

M2= 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 graphSE,

where V(SE) = Fq× Fq and for any two distinct vertices a,b∈V(SE),{a,b}is an edge ofSE if and

only if a2=b2 It follows that SE is a (q−1)-regular graph Since SPq(λ) is a q-regular graph, q

is an eigenvalue of M with the all-one eigenvector 1 The graph SPq(λ)is connected, therefore the

eigenvalue q has multiplicity one Similar to the proof ofTheorem 3.1, for any other eigenvalueθ of

SPq,θ2<2q−1 The theorem follows 2

We have an immediate corollary ofTheorem 2.1andTheorem 3.3 Note that, our result also follows from[4, Theorem 4.1]

Corollary 3.4 Let U,V ⊂ Fq× Fq For anyλ ∈ Fq , the number of solutions of the equation

x1+x2+x3x4= λ, (x1,x3)∈U, (x2,x4)∈V

is

S= |U||V|

q + θ q|U||V|,

whereθ2<2.

Taking U=V =A×A for a large subset A of the fieldFq, 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 ∈ F×

q, the product-sum graphPSq(λ)is defined as follows The vertex set of the prod-uct graph PSq(λ) is the set F×

q × Fq Two vertices a= (a1,a2) and b= (b1,b2) ∈V(PSq(λ)) are connected by an edge, {a,b} ∈E(PSq(λ)), if and only if a1b1(a2+b2) = λ We have the following pseudo-randomness of the product graphPSq(λ)

Theorem 3.6 The graphPSq(λ)is an



(q −1)q,q −1, 

3q

-graph.

Proof It is clear that PSq(λ) is a regular graph of order(q−1)q and valency q−1 We now

esti-mate the eigenvalues of this multigraph (i.e graph with loops) For any a= (a1,a2) =b= (b1,b2) ∈

V(PSq(λ)), we count the number of solutions of the following system

a1x1(a2+x2)=b1x1(b2+x2)= λ, x= (x1,x 2)∈V

PSq(λ) 

.

The system has a unique solution

x1= λ(b1−a1)

(a2−b2)a1b1,

x2=a1a2−b1b2

b1−a1

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 ofPSq(λ) It follows that

where J is the all-one matrix, I is the identity matrix, and E is the adjacency matrix of the graph

SE, where V( SE) = F×

q × Fq and for any two distinct vertices a,b∈V( SE), {a,b} is an edge ofSE

if and only if a1=b1 or a2=b2 It follows that SE is a(2q−3)-regular graph Since PSq(λ) is a

(q−1)-regular graph,(q−1)is an eigenvalue of M with the all-one eigenvector 1 The graphPSq(λ)

is connected, therefore the eigenvalue q−1 has multiplicity one It is clear that PSq(λ) contains (many) triangles which implies that the graph is not bipartite Hence, for any other eigenvalueθ of

PSq(λ),|θ| <q−1 Let v θ denote the corresponding eigenvector ofθ Note that v θ∈1⊥, so J v

θ=0

It follows from(3.2)that(θ2−q+2)v θ= −E v θ SinceSEis a 2(q−1)-regular graph, absolute values

of all eigenvalues ofSE are at most 2(q−1) This implies thatθ22q−3+q−2<3q The theorem

follows 2

We have an immediate corollary ofTheorem 2.1andTheorem 3.6

Corollary 3.7 Let U,V⊂ F×

q × Fq For anyλ ∈ F×

q , the number of solutions of the equation

x1x2(x3+x4)= λ, (x1,x3)∈U, (x2,x 4)∈V

is

S= |U||V|

q + θ q|U||V|,

whereθ2<3.

Taking U=V=A×A for a large subset A of the fieldFq, we conclude that the product of the sum set A+A and the product set A·A containsF×

q

Corollary 3.8 For any A⊂ Fq of cardinality|A| > √3q34, thenF×

q ⊂A·A· (A+A).

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