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Ed. by Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna


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Volume 31, Issue 1

Issues

Expansion for the product of matrices in groups

Doowon Koh / Thang Pham / Chun-Yen Shen / Anh Vinh Le
Published Online: 2018-08-18 | DOI: https://doi.org/10.1515/forum-2018-0063

Abstract

In this paper, we give strong lower bounds on the size of the sets of products of matrices in some certain groups. More precisely, we prove an analogue of a result due to Chapman and Iosevich for matrices in SL2(𝔽p) with restricted entries on a small set. We also provide extensions of some recent results on expansion for cubes in Heisenberg group due to Hegyvári and Hennecart.

Keywords: Matrix; finite fields; expanders

MSC 2010: 11B75; 20G40

1 Introduction

Let 𝔽p be a prime field. We denote by SL2(𝔽p) the set of 2×2 matrices with determinant one over 𝔽p. Given A𝔽p, we define

R(A):={(a11a12a21a22)SL2(𝔽p):a11,a12,a21A}.

For any two sets X and Y of matrices, we define XY:={xy:xX,yY}. It was proved by Chapman and Iosevich [1] by Fourier analytic methods that if |A|p56, then

|R(A)R(A)|p3.

Throughout this paper, the notation UV means UcV for some absolute constant c>0, and UV means U(logU)-cV for some absolute constant c>0. It has been extensively studied about the size of the products of R(A). In particular, the breakthrough work of H. A. Helfgott [5] asserts that if E is a subset of SL2(𝔽p) and is not contained in any proper subgroup with |E|<p3-δ, then |EEE|>c|E|1+ϵ for some ϵ=ϵ(δ)>0. The result mentioned above, by Chapman and Iosevich, is to give a quantitative estimate when the size of the set A is large. However, it is considered to be a difficult problem to obtain some quantitative estimate for the same problem when the size of the set A is not large. It is basically because the Fourier analytic methods are effective only when the size of the set A is large. In this paper, we address the case of small sets, and give a lower bound on the size of R(A)R(A). Our first result is as follows.

Theorem 1.1.

Let AFp. If |A|cp1219 for some small constant c>0, then

|R(A)R(A)||A|72+112=|R(A)|76+136.

Remark 1.1.

We note that the exponent 76 in Theorem 1.1 can be easily obtained by using the estimate |AA+AA||A|32 given in [10]. Indeed, one can check that

|R(A)R(A)||AA+AA||A|2|A|2+32=|A|72=|R(A)|76.

However, in order to improve the exponent 72, we need to prove some more results on sum-product type problems (see Section 2 for more details).

Let 𝔽p be a prime field. For an integer n1, the Heisenberg group of degree n, denoted by 𝐇n(𝔽p), is defined by a set of the following matrices:

[𝐱,𝐲,z]:=[1𝐱z𝟎In𝐲𝐭0𝟎1],

where 𝐱,𝐲𝔽pn, z𝔽p, 𝐲𝐭 denotes the column vector of 𝐲, and In is the n×n identity matrix. For A,B,C𝔽p, we define

[An,Bn,C]:={[𝐱,𝐲,z]:𝐱An,𝐲Bn,zC}.

A similar question in the setting of the Heisenberg group over prime fields has been recently investigated by Hegyvári and Hennecart in [3], namely, they proved the following theorem.

Theorem 1.2 (Hegyvári–Hennecart, [3]).

For every ϵ>0, there exists a positive integer n0=n0(ϵ) such that if nn0, and [An,Bn,C]Hn(Fp) with

|[An,Bn,C]|>|𝐇n((𝔽p))|34+ϵ,

then there exists a nontrivial subgroup G of Hn(Fp) such that [An,Bn,C][An,Bn,C] contains at least

|[An,Bn,C]|p

cosets of G.

In a very recent paper, using results on sum-product estimates, Hegyvári and Hennecart [4] established some results in the case n=1. In particular, they proved that if A𝔽p with |A|p12, then

|[A,A,0][A,A,0]|min{p12|[A,A,0]|54,p-12|[A,A,0]|2}.

In the case when |A|p23, they also showed that

|[A,A,0][A,A,0]||[A,A,0]|74.

In this paper, we also extend this result to the setting of Heisenberg group of degree two. For simplicity, we write [A2,A2,0]2 and [A2,A2,A]2 for the products [A2,A2,0][A2,A2,0] and [A2,A2,A][A2,A2,A], respectively. We have the following theorems.

Theorem 1.3.

If AFp with |A|p12, then we have

|[A2,A2,0]2||A|112+25262=|[A2,A2,0]|118+251048.

Theorem 1.4.

Let AFp with |A|p916. Then we have

|[A2,A2,A]2||A|112+2390=|[A2,A2,A]|1110+23450.

The rest of this paper is organized to provide the complete proofs of our main theorems. More precisely, in Section 2 we give the proof of Theorem 1.1, and in Section 3 we complete the proofs of Theorems 1.3 and 1.4.

2 Proof of Theorem 1.1

In this section, without loss of generality, we assume that 0A. To prove Theorem 1.1, we need the following lemmas.

Lemma 2.1 ([9, Corollary 3.1]).

Let X,AFp with |X||A|. Then we have

|X+AA|min{|X|12|A|,p}.

Lemma 2.2.

Let AFp with |A|cp23 for a sufficiently small c>0. Then the number of tuples (a1,a2,a3,a4,a1,a2,a3,a4)A8 satisfying

a1a2+a3a4=a1a2+a3a4

is |A|132.

Proof.

For λ,β𝔽p{0}, one can follow the proof of [10, Theorem 3] to prove that the number of tuples (a1,a2,a3,a1,a2,a3)A6 such that a1a2+λa3=a1a2+βa3 is |A|92. Thus, we see that for each fixed pair (a4,a4)A2 the number of tuples (a1,a2,a3,a4,a1,a2,a3,a4)A8 satisfying

a1a2+a3a4=a1a2+a3a4

is |A|92. Taking the sum over all pairs (a4,a4)A2, the lemma follows. ∎

Lemma 2.3 ([15, Theorem 4]).

Let A,BFp with |A||B|, and let L be a finite set of lines in Fp2. Suppose that |A||B|2|L|3 and |A||L|p2. Then the number of incidences between A×B and lines in L, denoted by I(A×B,L), satisfies

I(A×B,L)|A|34|B|12|L|34+|L|.

The following is an improvement of [13, Lemma 23].

Lemma 2.4.

Let A,BFp. Then if |A|=|B|, and |A|2|AB|p2, we have

|A(B+x)||A|-12|AB|54

for any x0.

Proof.

It is clear that

|A(B+x)|1|A||B||{(u,u*,a,b)AB×AB×A×B:ub-1-u*a-1=x}|.

The number of such tuples (u,u*,a,b) is bounded by the number of incidences between points in A-1×AB and a set L of lines of the form b-1Y-u*X=x with bB, u*AB. Notice that |A|=|A-1| and |L|=|B||AB|. Thus, if |A|=|B| and |A|2|AB|p2, Lemma 2.3 implies that

I(A-1×AB,L)|A|32|AB|54,

which completes proof of the theorem. ∎

Lemma 2.5 ([8, Theorem 2]).

If AFp with |A|p916, then we have

|A±A|18|AA|9|A|32.

We are now ready to prove Theorem 1.1.

Proof of Theorem 1.1.

Without loss of generality, we may assume that 0A. Let M1 and M2 be matrices in R(A) presented as follows:

M1:=(a11a12a211+a12a21a11),M2:=(b11b12b211+b12b21b11).

Suppose that

M1M2=(tαβ1+αβt),

where t0 and α,β𝔽p. Then we have the following system

a11b11+a12b21=t,b12tb11+a12b11=α,a21ta11+b21a11=β.(2.1)

Let us identify the matrix M1M2 with

(t,α,β)𝔽p*×𝔽p2.

Notice that R(A)R(A) contains each (t,α,β)𝔽p*×𝔽p2 satisfying system (2.1) for some (a11,a12,a21,b11,b12,b21) in A6. Therefore, we aim to estimate a lower bound of the number of (t,α,β)𝔽p*×𝔽p2 such that system (2.1) holds for some (a11,a12,a21,b11,b12,b21)A6. To this end, let ϵ>0 be a parameter chosen later. We now consider the following two cases:

Case 1. If |AA||A|1+ϵ, then it follows from Lemma 2.1 that

|AA+AA|min{|A|32+ϵ2,p}=|A|32+ϵ2,

where we assume that |A|p23+ϵ. From system (2.1) and the above fact, we obtain that if |A|p23+ϵ and |AA||A|1+ϵ, then

|R(A)R(A)||AA+AA||A|2|A|72+ϵ2,(2.2)

where the first follows, because in system (2.1), for each nonzero tAA+AA, if we fix a quadruple (a11,b11,a12,b21)A4 with a11b11+a12b21=t, then α,β are determined in terms of b12A and a21A, respectively.

Case 2. If |AA||A|1+ϵ, then we consider as follows. For t,α,β𝔽p, let ν(t,α,β) be the number of solutions (a11,a12,a21,b11,b12,b21) of system (2.1). For the case t=0, we have

α,βν(0,α,β)|A|5.

Indeed, for each choice of (b11,a12,b21)A3, a11 is determined uniquely, and α,β are determined. In addition, a21 and b12 can be taken as arbitrary elements of A.

By the Cauchy–Schwarz inequality, we have

(|A|6-|A|5)2(t0,α,βν(t,α,β))2|R(A)R(A)|t0,α,βν2(t,α,β).

This implies that

|R(A)R(A)||A|12T,(2.3)

where T:=t0,α,βν2(t,α,β).

In the next step, we are going to show that

T|A|8+5ϵ2.

To see this, observe by definition of ν(t,α,β) that for each (t,α,β)𝔽p*×𝔽p2, the value ν2(t,α,β) is the number of 12-tuples (a11,a12,a21,b11,b12,b21,a11,a12,a21,b11,b12,b21)A12 satisfying the following:

a11b11+a12b21=t=a11b11+a12b21,b12tb11+a12b11=α=b12tb11+a12b11,a21ta11+b21a11=β=a21ta11+b21a11.

Thus the value of T=t0,α,βν2(t,α,β) can be written as t0Ω(t), where Ω(t) denotes the number of 12-tuples (a11,a12,a21,b11,b12,b21,a11,a12,a21,b11,b12,b21)A12 satisfying the following:

a11b11+a12b21=t=a11b11+a12b21,(2.4)b12tb11+a12b11=b12tb11+a12b11,(2.5)a21ta11+b21a11=a21ta11+b21a11.(2.6)

Now notice that Lemma 2.2 implies that if |A|p23, then there are at most |A|132 8-tuples (a11,b11,a12,b21,a11,b11,a12,b21) in A8 satisfying equations (2.4) for some t0. One can also check that among these tuples, there are at most |A|6 (12|A|132) tuples with a12b11-1-a12b11-1=0. Hence, without loss of generality, we may assume that all tuples satisfy a12b11-1-a12b11-10.

For such a fixed 8-tuple (a11,b11,a12,b21,a11,b11,a12,b21)A8, we now deal with equation (2.5) which can be rewritten as

b12t-1b11+a12b11-1=b12t-1b11+a12b11-1.(2.7)

Set

Q=tb11A,Q=tb11Aandx=a12b11-1-a12b11-10.

Then the number of solutions (b12,b12)A2 of (2.7) is the size of Q(Q-x). It is clear that |Q|=|Q|=|A|, because t0, and we have assumed that 0A so that tb11,tb110. We also see that

|Q|2|QQ|=|A|2|AA||A|3+ϵ,

where we used the assumption that |AA||A|1+ϵ. Applying Lemma 2.4, we obtain that if |A|p23+ϵ, then

|Q(Q-x)||A|34+5ϵ4.

The same argument works identically for equation (2.6) which can be restated as

a21t-1a11+b21a11-1=a21t-1a11+b21a11-1.

In short, we have proved that if |A|p23+ϵ and |AA||A|1+ϵ, then

T|A|132|A|34+5ϵ4|A|34+5ϵ4=|A|8+5ϵ2.

Therefore, combining (2.3) with this estimate yields that if |A|p23+ϵ and |AA||A|1+ϵ, then

|R(A)R(A)||A|4-5ϵ2.(2.8)

Finally, if we choose ϵ=16, then it follows from (2.2) and (2.8) that if |A|p1219, then

|R(A)R(A)||A|72+112,

which completes the proof of Theorem 1.1. ∎

In the case of arbitrary finite fields, we have the following result.

Theorem 2.6.

Let q=pn and let A be a subset of Fq*. If |AλF||F|12 for any proper subfield F of Fq and any λFq, then we have

|R(A)R(A)||A|3+15=|R(A)|1+115.

To prove Theorem 2.6, we make use of the following results.

Theorem 2.7 ([7]).

With the assumptions of Theorem 2.6, we have

max{|A+A|,|AA|}|A|1211.

Theorem 2.8.

For A,BFq, suppose that |AλF|,|BλF||F|12 for any subfield F of Fq and any λFq. Then we have

|A+AB|min{|A||B|15,|A|34|B|24}.

Corollary 2.9.

For AFq, suppose that |AλF||F|12 for any subfield F of Fq and any λFq. Then we have

|AA+AA||A|65.

Proof.

Given a nonzero xA, we have

|AA+AA|=|AxAx+AxAx||AxAx+Ax||A|65

by Theorem 2.8. ∎

We are now ready to prove Theorem 2.6.

Proof of Theorem 2.6.

Recall from (2.2) that

|R(A)R(A)||AA+AA||A|2.

Thus the theorem follows directly from Corollary 2.9. ∎

In the rest of this section, we present the proof of Theorem 2.8, for which the authors communicated with Oliver Roche-Newton.

2.1 Proof of Theorem 2.8

To prove Theorem 2.8, we make use of the following lemmas.

The first lemma is the Plünnecke–Ruzsa inequality.

Lemma 2.10 ([11, Theorems 1.6.1 and 1.81]).

Let X,B1,,Bk be subsets of Fq. Then we have

|B1++Bk||X+B1||X+Bk||X|k-1𝑎𝑛𝑑|B1-B2||X+B1||X+B2||X|.

One can modify the proof of [6, Corollary 1.5] due to Katz and Shen to obtain the following:

Lemma 2.11.

Let X,B1,,Bk be subsets in Fq. Then for any 0<ϵ<1, there exists a subset XX such that |X|(1-ϵ)|X| and

|X+B1++Bk|c|X+B1||X+Bk||X|k-1

for some positive constant c=c(ϵ).

We also have the following lemma from [7].

Lemma 2.12.

Let B be a subset of Fq with at least two elements, and define FB as the subfield generated by B. Then there exists a polynomial P(x1,,xn) of several variables with coefficients belonging to the prime field Fp such that

P(B,,B)=𝔽B.

We are now ready to prove Theorem 2.8.

Proof of Theorem 2.8.

Without loss of generality, we may assume 1A by considering 1xA for some xA. We first define the ratio set

R(A,B):={a1-a2b1-b2:a1,a2A,b1,b2B,b1b2}.

We now consider the following cases.

Case 1: 1+R(A,B)R(A,B). In this case, there exist a1,a2A and b1b2B such that

r:=1+a1-a2b1-b2R(A,B).

First, we apply Lemma 2.11 so that there exists a subset AA such that |A||A| and

|(b1-b2)A+(b1-b2)B+(a1-a2)B||A+B||(b1-b2)A+(a1-a2)B||A|.(2.9)

On the other hand, we have

|(b1-b2)A+(b1-b2)B+(a1-a2)B||A+rB|.(2.10)

Since rR(A,B), the equation a1+rb1=a2+rb2 has no nontrivial solutions, i.e. solutions (a1,b1,a2,b2) with b1b2. This implies that

|A+rB|=|A||B|.(2.11)

We now give an upper bound for (b1-b2)A+(a1-a2)B=b1A+a1B-b2A-a2B which will be used in the rest of the proof.

Lemma 2.11 tells us that there exists a subset XA such that |X||A| and

|X+b1A+a1B||A+b1A||A+a1B||A||A+AB|2|A|,

and there exists a subset XX with |X||X| such that

|X+b2A+a2B||X+b2A||X+a2B||X||A+AB|2|A|.

Applying Lemma 2.10, we have

|b1A+a1B-b2A-a2B||X+b1A+a1B||X+b2A+a2B||X||X+b1A+a1B||X+b2A+a2B||A||A+BA|4|A|3.(2.12)

Putting (2.9)–(2.12) together, we obtain

|A+AB|5|A|5|B|,

and we are done.

Case 2: BR(A,B)R(A,B). Similarly, in this case, there exist a1,a2A and b,b1,b2B such that

r:=ba1-a2b1-b2R(A,B).

Since 0R(A,B), we have b0 and a1a2. This gives us that r-1 exists.

Using the same argument as above, we have

|A||B|=|A+rB|=|r-1A+B||b-1A+A||(a1-a2)B+(b1-b2)A||A||A+AB||A+AB|4|A|4.

Thus, we obtain

|A+AB|5|A|5|B|,

and we are done.

Case 3: B-1R(A,B)R(A,B). As above, in this case, there exist a1,a2A and b0, b1b2B such that

r:=b-1a1-a2b1-b2R(A,B).

Since 0R(A,B), we have a1a2. This gives us that r-1 exists. The rest is the same as Case 2.

Case 4. We consider the case when

1+R(A,B)R(A,B),BR(A,B)R(A,B),B-1R(A,B)R(A,B).

Now we are going to show that for any polynomial P(x1,,xn) in n variables, for some positive integer n, and coefficients belonging in 𝔽p such that

P(B,,B)+R(A,B)R(A,B).

Indeed, it is enough to show that

1+R(A,B)R(A,B),Bd+R(A,B)R(A,B)

for any integer d1 and Bd=BB (d times).

It follows from the assumption that the first condition 1+R(A,B)R(A,B) is satisfied. For the second condition, it is sufficient to prove it for d=2, since we can use inductive arguments.

Let b,b be arbitrary elements in B. We now show that

bb+R(A,B)R(A,B).

If either b=0 or b=0, then we are done. Thus, we may assume that b0 and b0.

First, we have

b+R(A,B)=b(1+b-1R(A,B))b(1+R(A,B)))R(A,B)

and

bb+R(A,B)=b(b+b-1R(A,B))b(b+R(A,B))bR(A,B)R(A,B).

In other words, for any polynomial P(x1,x2,,xn)𝔽p[x1,,xn], we have

P(B,,B)+R(A,B)R(A,B).

Furthermore, Lemma 2.12 tells us that there exists a polynomial P such that P(B,,B)=𝔽B. This implies that

𝔽B+R(A,B)R(A,B).

It follows from our assumption of the theorem that

|B|=|B𝔽B||𝔽B|12.

Hence, |R(A,B)||𝔽B||B|2.

Next, we shall show that there exists rR(A,B) such that either

|A+rB||A||B|or|A+rB||B|2.

Indeed, let E+(X,Y) be the number of tuples (x1,x2,y1,y2)X2×Y2 such that

x1+y1=x2+y2.

We have that the sum rR(A,B)E+(A,rB) is the number of tuples (a1,a2,b1,b2)A2×B2 such that

a1+rb1=a2+rb2

with a1,a2A, b1,b2B and rR(A,B). It is easy to see that there are at most |R(A,B)||A||B| tuples with a1=a2, b1=b2, and at most |A|2|B|2 tuples with b1b2. Therefore, we get

rR(A,B)E+(A,rB)|R(A,B)||A||B|+|A|2|B|2|R(A,B)|(|A||B|+|A|2).

By the pigeon-hole principle, there exists r:=a1-a2b1-b2R(A,B) such that

E(A,rB)|A||B|+|A|2.

So, either

|A+rB||A||B|or|A+rB||B|2.

We now fall into two small cases:

  • (1)

    If |A+rB||A||B|, then, applying Lemma 2.10, we have

    |A||B|=|A+rB|=|(b1-b2)A+(a1-a2)B||AB+A|4|A|3,

    which gives us

    |A+AB||A||B|12.

  • (2)

    If |A+rB||B|2, then we have

    |B|2|A+rB|=|(b1-b2)A+(a1-a2)B||AB+A|4|A|3,

    which gives us

    |A+AB||A|34|B|24.

This completes the proof of the theorem. ∎

3 Proofs of Theorems 1.3 and 1.4

In the proof of Theorem 1.3, we make use of the following version of the Balog–Szemerédi–Gowers theorem due to Schoen [12].

Theorem 3.1 ([12, Theorem 1.1]).

Let G be an Abelian group. Suppose that A is a subset of G, and E+(A) denotes the additive energy which is the number of solutions (a,b,c,d)A4 to the equation a+b=c+d. If E+(A) is equal to k|A|3, then there exists AA with |A|k|A| such that

|A-A|k-4|A|.

We will also need the following results.

Theorem 3.2.

For A,B,C,DFp, let Q(A,B,C,D) be the number of 8-tuples

(a1,b1,c1,d1,a2,b2,c2,d2)(A×B×C×D)2

such that

a1b1+c1d1=a2b2+c2d2.

We have

Q(A,B,C,D)|A|2|B|2|C|2|D|2p+|C|2|B||D|32|A|12E×(A,B)12+|A||D|3|B||C|+|A|3|D||B||C|,

where

E×(A,B)=#{(a1,a2,b1,b2)A2×B2:a1b1=a2b2}.

To prove this theorem, we need the following version of the point-plane incidence bound due to Rudnev in [8].

Theorem 3.3 ([8]).

Let P be a set of points in Fp3 and let Π be a set of planes in Fp3. Suppose that |P||Π|, and there are at most k collinear points in P for some k. Then the number of incidences between P and Π is bounded by

I(P,Π)|P||Π|p+|P|12|Π|+k|P|.

We are now ready to prove Theorem 3.2. We will follow the ideas of [14, proof of Theorem 32].

Proof of Theorem 3.2.

We have

Q(A,B,C,D)=λ,μrCD(λ)rAB(μ)n(λ,μ),

where rCD(λ) is the number of pairs (c,d)C×D such that cd=λ, rAB(μ) is the number of pairs (a,b)A×B such that ab=μ, and n(λ,μ)=xrAB+λ(x)rCD+μ(x). If we split the sum Q(A,B,C,D) into intervals, we get

Q(A,B,C,D)i=1L1j=1L2λ,μn(λ,μ)rCD(i)(λ)rAB(j)(μ),

where L1log(|C||D|), L2log(|A||B|), rAB(i)(μ) is the restriction of the function rAB(x) on the set

Pi:={μ:ΔirAB(μ)<2Δi},

and rCD(i)(λ) is the restriction of the function rCD(x) on the set

Pi:={λ:ΔirCD(λ)<2Δi}.

Applying the pigeon-hole principle twice, there exist sets Pi and Pj such that

Q(A,B,C,D)λ,μn(λ,μ)rCD(i)(λ)rAB(j)(μ)ΔiΔjλ,μn(λ,μ)Pi(μ)Pj(λ),

where Pi(x) is the indicator function of the set Pi. For simplicity, we suppose that i=1 and j=2.

One can check that the sum

λ,μn(λ,μ)P1(λ)P2(μ)

is the number of incidences between points (a,d,λ)A×D×P1𝔽p3 and planes in 𝔽p3 defined by

bX-cY+Z=μ,

where bB, cC, μP2.

With the way we define the plane set, it follows from [2] that we can apply Theorem 3.3 with

k=max{|A|,|D|}.

Thus, we obtain

Q(A,B,C,D)Δ1Δ2(|A||B||C||D||P1||P2|p)+Δ1Δ2(|A|12|B||C||D|12|P1|12|P2|+max{|A|,|D|}|A||D||P1|).(3.1)

It is clear in our argument that we can switch the point set and the plane set, we also can do the same thing for P1 and P2 in the definition of the point set and the plane set. So, without loss of generality, we can assume that |P1||P2|, |A||D||B||C|. We now consider the following cases:

  • If the second term dominates, then we have

    Q(A,B,C,D)|C|2|B||D|32|A|12E×(A,B)12,

    since

    Δ2|P2||C||D|,Δ1|P1|12E×(A,B)12.

  • If the first term dominates, then we have

    Q(A,B,C,D)|A|2|B|2|C|2|D|2p,

    since

    Δ2|P2||C||D|,Δ1|P1||A||B|.

  • If the last term dominates, then we study the following:

    • (1)

      Suppose |A||D|. If |D||P2|, then it is easy to check that the second term in (3.1) will be bigger than the last term. Thus, we can suppose that |D||P2|. Since |P1||P2|, we have

      |A||D|2|P1||A||D|3.

      On the other hand, it is clear that Δ1Δ2|B||C|. This means

      Q(A,B,C,D)|A||D|3|B||C|.

    • (2)

      Suppose |A||D|. By repeating the same argument, we obtain

      Q(A,B,C,D)|A|3|D||B||C|.

This completes the proof of the theorem. ∎

Proof of Theorem 1.3.

Let N be the number of tuples

(x1,y1,z1,t1,x2,y2,z2,t2,x1,y1,z1,t1,x2,y2,z2,t2)A16

such that [𝐱,𝐲,0][𝐳,𝐭,0]=[𝐱,𝐲,0][𝐳,𝐭,0]. This can be expressed as follows:

(1x1x20010y1001y20001)(1z1z20010t1001t20001)=(1x1x20010y1001y20001)(1z1z20010t1001t20001).(3.2)

Thus, by the Cauchy–Schwarz inequality, we have

|[A2,A2,0]2||A|16N.(3.3)

From (3.2), observe that N is the number of tuples

(x1,y1,z1,t1,x2,y2,z2,t2,x1,y1,z1,t1,x2,y2,z2,t2)A16

satisfying the following system:

x1+z1=x1+z1,x2+z2=x2+z2,(3.4)y1+t1=y1+t1,y2+t2=y2+t2,(3.5)x1t1+x2t2=x1t1+x2t2.(3.6)

We now insert new variables s1,s2,s3,s4A+A in (3.4) and (3.5) as follows:

x1+z1=s1=x1+z1,x2+z2=s2=x2+z2,y1+t1=s3=y1+t1,y2+t2=s4=y2+t2.

Set Asi=A(si-A), then we have x1,x1As1, x2,x2As1, y1,y1As3, y2,y2As4.

In this setting, we have

N=s1,s2,s3,s4A+AQ(As1,As2,As3,As4).

Applying Theorem 3.2, and assuming the second and third terms are larger than the first term, we have

Ns1,s2,s3,s4|As1|12|As2||As3|2|As4|32E×(As1,As2)12+s1,s2,s3,s4|As1||As2||As3||As4|3+s1,s2,s3,s4|As1|3|As2||As3||As4|(s1,s2|As1|12|As2|E×(As1,As2)12)(s3|As3|2)(s4|As4|32)+|A|10|A|E+(A)32(s1,s2|As1|12|As2|E×(As1,As2)12)+|A|10,

where we have used the fact that

s|As|32(s|As|)12(s|As|2)12,s|As|2E+(A).

Moreover, using the fact E×(A,B)|A|2|B|, we have

s1,s2|As1|12|As2|E×(As1,As2)12s1,s2|As1|32|As2|32|A|2E+(A,A).

In other words, we have proved that

N|A|3E+(A)52+|A|10.

If N|A|10, then the theorem follows from (3.3). Therefore, we can assume that

N|A|3E+(A)52.

Let ϵ be a parameter chosen later. We now consider two cases:

Case 1. Suppose that E+(A)<|A|3-ϵ. Then we have

N|A|92+6-5ϵ2.

From (3.3), this implies that

|[A2,A2,0]2||A|112+5ϵ2.

Case 2. Suppose that E+(A)|A|3-ϵ. Then we can write E+(A)=|A|3-ϵ for some ϵ<ϵ<1. Notice that Theorem 3.1 implies that there exists a subset AA such that |A||A|1-ϵ and

|A-A||A|4ϵ|A||A|1+4ϵ1-ϵ.

Since |A|p916 by our assumption, using Lemma 2.5 with the above inequality gives

|AA||A|149-8ϵ1-ϵ.(3.7)

Moreover, one can easily check that

|[A2,A2,0]2||A|4|{x1t1+x2t2:x1,t1,x2,t2A}|.

Therefore,

|[A2,A2,0]2||A|4|{x1t1+x2t2:x1,t1A,x2,t2A}|.

Moreover, Lemma 2.1 gives us

|{x1t1+x2t2:x1,t1A,x2,t2A}|min{|A||AA|12,p}min{|A|1+(1-ϵ)(79-4ϵ1-ϵ),p},

where we also utilized inequality (3.7) and the fact that |A||A|1-ϵ.

Thus, we obtain that if |A|p916, then

|[A2,A2,0]2|min{|A|5+(1-ϵ)(79-4ϵ1-ϵ),p|A|4}|A|5+(1-ϵ)(79-4ϵ1-ϵ)

provided that |A|1+(1-ϵ)(79-4ϵ1-ϵ)p. It is clear that with ϵ=5131, this estimate is satisfied since |A|p12. We also obtain

|[A2,A2,0]2||A|112+25262.

Finally, suppose the first term is the largest term. Then we have NE+(A,A)4p which is smaller than |A|12p. Thus (3.3) shows that

|[A2,A2,0]2|p|A|4|A|6.

This is more than what we want and completes the proof of the theorem. ∎

Over finite arbitrary fields 𝔽q, we have the following result.

Theorem 3.4.

Let q=pn and let A be a subset of Fq*. If |AλF||F|12 for any proper subfield F of Fq and any λFq, then we have

|[A,A,0]2||A|3+111,𝑎𝑛𝑑|[A2,A2,0]2||A|5+15.

Proof.

We first observe that

|[A,A,0]2||A|2max{|A+A|,|AA|}

and

|[A2,A2,0]2||A|4|AA+AA|.

It follows from Theorems 2.7 and 2.8 that

|AA+AA||A|65andmax{|A+A|,|AA|}|A|1211.

Therefore, we obtain

|[A,A,0]2||A|3+111and|[A2,A2,0]2||A|5+15,

which completes the proof of the theorem. ∎

Proof of Theorem 1.4.

One can observe that

|[A2,A2,A]2||A|4|AA+AA+A+A|.(3.8)

We now prove that if |A|p916, then

|AA+AA+A+A||A|7945.

Indeed, we first prove that if |A|p916, then

|AA+A+A||A|32+190.(3.9)

To prove this inequality, we consider the following cases:

Case 1. If |A+A||A|1+ϵ, then it follows from Lemma 2.1 that

|AA+A+A|min{|A|32+ϵ2,p}=|A|32+ϵ2,(3.10)

whenever |A|p23+ϵ.

Case 1. If |A+A||A|1+ϵ, then Lemma 2.5 gives us that |AA||A|149-2ϵ under the condition |A|p916. Hence, if |A|p916, then

|AA+A+A||AA||A|149-2ϵ.(3.11)

Choosing ϵ=145, we see from (3.10) and (3.11) that if |A|p4568 and |A|p916, then

|AA+A+A||A|32+190.

Since p4568p916, we establish inequality (3.9).

By Lemma 2.1 and inequality (3.9), we see that if |A|p916, then

|AA+(AA+A+A)|min{|A||AA+A+A|12,p}min{|A|74+1180,p}=|A|7945.

Finally, combining (3.8) and this estimate, we conclude that if |A|p916, then

|[A2,A2,A]2||A|112+2390,

which completes the proof of Theorem 1.4. ∎

Acknowledgements

The authors would like to deeply thank Oliver Roche-Newton and Ilya Shkredov for many helpful discussions that make nice improvement for our Theorem 1.3. The authors would like to thank the referee for valuable suggestions.

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About the article


Received: 2018-03-09

Revised: 2018-07-05

Published Online: 2018-08-18

Published in Print: 2019-01-01


Funding Source: National Research Foundation of Korea

Award identifier / Grant number: NRF-2015R1A1A1A05001374

Funding Source: Schweizerischer Nationalfonds zur Förderung der Wissenschaftlichen Forschung

Award identifier / Grant number: P2ELP2175050

Funding Source: Ministry of Science and Technology, Taiwan

Award identifier / Grant number: 104-2628-M-002-015-MY4

D. Koh was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2015R1A1A1A05001374). T. Pham was supported by Swiss National Science Foundation grant P2ELP2175050. C.-Y. Shen was supported in part by MOST, through grant 104-2628-M-002-015-MY4.


Citation Information: Forum Mathematicum, Volume 31, Issue 1, Pages 35–48, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2018-0063.

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