Fourier analysis and expanding phenomena in finite fields

In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi, Vinh and Vu using spectral graph theory. In addition, several generalizations of these results are given. In the case that $A$ is a subset of a prime field $\mathbb F_p$ of size less than $p^{1/2}$ it is shown that $|\{a^2+b:a,b \in A\}|\geq C |A|^{147/146}$, where $|\cdot|$ denotes the cardinality of the set and $C$ is an absolute constant.


introduction
Let F be a field and E be a finite subset of F d , the d-dimensional vector space over F. Given a function f : the image of f under the subset E. We shall say that f is a d-variable expander with expansion index ǫ if |f (E)| ≥ C ǫ |E| 1 /d+ǫ , for every subset E possibly under some general density or structural assumptions on E.
Several classical problems in additive and geometric combinatorics deal with showing that certain polynomials have the expander property. Given a finite subset E ⊂ R d the Erdős distance problem deals with the case of ∆ : that is ∆ is a 2d-variable expander with expansion index 1/2d. Taking E to be a piece of the integer lattice shows that one cannot in general do better. (Throughout the paper we will write X Y to mean X ≤ CY where C is a universal constant, which may vary from line to line but are always universal. It is also clear that when the quantities X, Y have f (A) involved for some polynomial f , the implied constant may also depend on the degree of f . In addition, we will write X Y in the case that for every δ > 0 there exists C δ > 0 such that X ≤ C δ t δ Y where t is a large controlling parameter.) The fact that ∆ is an expander goes back to the original 1945 by Erdős ([9]). The best results on the problem in two dimensions, due to Katz and Tardos ([21]), are based on a previous breakthrough by Solymosi and Toth ([35]). For the best known results in higher dimensions see [36] and [37].
The so-called sum and product problems may also be rephrased as results about expanders. Specifically, dealing with the fact that for a given set if one function is non-expanding then it may imply that another function is an expander.
By far the most studied case is that of d = 2. In this case Erdős and Szemerédi gave the bound max(|A + A|, |A · A|) |A| 1+ǫ , for a small but positive ǫ.
Explicit bounds on ǫ where ǫ ≥ 1/31 was given by Nathanson ([25]) and ǫ ≥ 1/15 by Ford ([11]). A breakthrough by Elekes ( [7]) connected the problem to incidence geometry applying the Szeméredi-Trotter incidence theorem giving ǫ ≥ 1/4. This was improved by Solymosi ([32]) to ǫ ≥ 3/14 − δ where δ → 0 as |A| → ∞. These bounds hold in the more general context of finite subsets of R. In this case the best known bound, due to Solymosi ([34]), is given by max(|A + A|, |A · A|) |A| With regards to the general conjecture much less is known. However, Bourgain and Chang ( [2]) showed that if A is a subset of Z then for any n ∈ N, there exists The sum-product problems have been explored in the context of a variety of rings. In this paper we will be concerned with subsets of the finite fields F q . In this context the situation appears to be more complicated due to the fact that one may not rely on the topological properties of the real numbers. It is known, however, via ground breaking work in [4] that if A ⊂ F p , p a prime, and if |A| ≤ p 1−ǫ for some ǫ > 0, then there exists δ > 0 such that max(|A + A|, |A · A|) |A| 1+δ .
This bound was given via combinatorial means and did not yield a precise relationship between δ and ǫ. In [18] the first listed author along with Iosevich and Solymosi, used Fourier analysis to develop incidence theory between points and hyperbolas in F 2 q , the 2-dimensional vector space over F q . This led to for the first time, a concrete value of δ, for |A| > q 1/2 . This bound on |A| is natural in finite fields which are not necessarily prime fields where subfields of size q 1/2 give the trivial bound. Garaev ([13]) applied a method of Elekes ( [7]) to give the bound max(|A + A|, |A · A|) min(|A| 1/2 q 1/2 , |A| 2 q −1/2 ).
Solymosi ([33]) applied spectral graph theory to give a similar bound for a general class of functions f of which polynomials of integer coefficients and degrees greater than one are members. Let A, B, C be subsets of F q . Then max(|A + B|, |f (A) + C|) min(|A| 1/2 q 1/2 , |A||B| 1/2 |C| 1/2 q −1/2 ).
Setting B = f (A) and C = A immediately gives the expander This result is analogous to the work done by Elekes, Nathanson and Ruzsa ( [8]) in the real numbers.
In the case of prime fields and f (x 1 , x 2 ) = x 1 x 2 , Garaev ( [12]) used combinatorial methods to give non-trivial bounds for all ranges of |A|. In the case of |A| ≤ p 1/2 then the best result currently is due to the second listed author ( [23]) giving max(|A + A|, |A · A|) |A| 13/12 , slightly improving the result of the third listed author in [30].
In prime fields it follows from the result of Glibichuk and Konyagin ( [15]) that for |A| ≤ p 1/2 one has that |A · A + A| |A| 7/6 . This shows that under these constraints one has that f (x 1 , x 2 , x 3 ) = x 1 x 2 + x 3 is a three-variable expander of expansion index 1/18. Bourgain ([1]) answered a question of Widgerson giving examples of twovariable expanders. Specifically, showing that f (x 1 , x 2 ) = x 1 (x 1 +x 2 ) and g(x 1 , x 2 ) = x 1 (x 2 +1) are expanders. However, Bourgain did not give explicit expansion indexes. In the later case Garaev and the third listed author ( [14]) showed that the expansion index could be taken to be 1/210 − o(1).
Given the nature of an expander it seems that the image of a sufficiently large set should expand enough to encapsulate most if not all of a finite field. However, some expanders exhibit a stronger version of this property than others. It seems expanders fall into one of the following three types.
be a given function.
• We say that f is a strong expander if there exists an ǫ > 0 such that for all |A| q 1−ǫ one has that |f (A, . . . , A)| ≥ q − k for a fixed constant k. • We say that f is a moderate expander if there exists an ǫ > 0 such that for |A| q 1−ǫ one has that |f (A, . . . , A)| q. • We say that f is a weak expander if there exist an ǫ > 0 and a δ < 1 such that for all |A| q 1−ǫ one has that |f (A, . . . , A)| |A| δ q 1−δ .
An interesting question is to determine the minimal number of variables an expander of a certain type can have. As shown in [16] it is impossible for a two-variable expander to be a strong expander. It is unknown as to whether there are moderate expanders which are not strong expanders for some ǫ.

An example of a strong expander is the function
is a moderate expander for |A| q 3/4 . A result of Garaev and the third listed author ( [14]) shows that f (x 1 , x 2 ) = x 1 (x 2 + 1) is a weak expander for |A| q 2/3 .

Sum-product estimates
q } and the group operation ⊙ is inherited from each coordinate group. Define the Fourier transform of any given function f : where χ = (χ 1 , . . . , χ d ) and χ j denotes the additive or multiplicative character corresponding to G j and by the the function χ(x) we mean We also define the convolution of functions f, g by where y −1 is the inverse of y in G d . Then the following are easy to verify: Define ( [39]) the uniformity norm (or Fourier bias) of f by where by χ 0 we mean (χ 0 1 , . . . , χ 0 d ) for χ 0 j the trivial character of the coordinate group. We first give a modified version of a lemma of Solymosi ([33]).
which in turn by Cauchy-Schwarz and Plancherel is ≤ |G d | X u |Y ||P |.
We say that a set F is Salem with constant C if This gives the following theorem.
Remark 2.3. This theorem can be viewed as a finite field version of the main theorem in [8] by Elekes, Nathanson and Ruzsa, in which the authors investigated the incidences between points and convex curves in the real plane, and applied the incidence bound to show |S + T | min(|S||T |, |S| 3/2 |T | 1/2 ) for any finite subset S of a strictly convex curve in R 2 , while T is arbitrary.
2.1. Salem Sets. Let F q be a finite field with characteristic p, and T r : F q → F p be the absolute trace function. It is well-known ( [24]) that the function χ defined by is a character of the additive group of F q , and every additive character χ of F q is of the form χ(c) = χ(bc) for some b ∈ F q . Note also the group of multiplicative characters of F q is a cyclic group. Denote by N(f ) the number of distinct roots of f ∈ F q [x] in its splitting field over F q . Then it is easy to see that The classical bound due to Weil as well as its generalization for mixed character sums may be used to show that certain setsX defined by polynomials are Salem.
Theorem 2.4 (Weil's Bound [24,26]). Let χ be a non-trivial additive character of F q and ψ be a non-trivial multiplicative character of F q of order s.
(2) Suppose that g ∈ F q [x] is not, up to a nonzero multiplicative constant, an s-th power of a polynomial in F q [x]. Then for any where d is the number of distinct roots of g in its splitting field over F q .
Particularly, taking f to be some constant function we get Suppose f contains some irreducible factors that are not factors of g such that the great common divisor of the powers of these factors in the canonical factorization of f is 1, and vice versa.
Then F is a Salem set with constant M.

Vu's non-degenerate polynomials. We give a generalization of Vu's result ([41]) using Theorem 2.2. Following Vu, a polynomial
and L is a linear form in x 1 , x 2 . We first recall the Schwarz-Zipple lemma ( [39]) and the Katz theorem in [22]. Theorem 2.8 (Katz). Let P (x 1 , x 2 ) be a polynomial of degree k in F 2 q which does not contain a linear factor. Let P −1 = {(x, y) ∈ F 2 q : P (x, y) = 0}. Then P −1 u k 2 q −3/2 , that is to say P −1 is a Salem set with respect to F 2 q . Theorem 2.9. Let P be a non-degenerate polynomial of degree k in F q [x 1 , x 2 ]. Then for any E, F ⊂ F 2 q with |E| ≫ k 2 q we have Proof. For each a ∈ F q , let P −1 (a) = {(x 1 , x 2 ) ∈ F 2 q : P (x 1 , x 2 ) = a}. By Vu's Lemma 5.1 ( [41]), there are at least q − (k − 1) elements a i such that P − a i does not contain a linear factor. We call such a i good and form the bad elements into a set ∆. By Lemma 2.7, for each z ∈ ∆ one has |P −1 (z)| ≤ kq. Hence z∈∆ |P −1 (z)| ≤ (k − 1)kq, and considering that |E| ≫ k 2 q we get E\ z∈∆ P −1 (z) ∼ |E|. Therefore, where M = max a∈∆ c |E ∩ P −1 (a)|. Now choose one a ∈ ∆ c which achieves the above maximum and define Combining Lemma 2.7, Theorem 2.8 with the deduction of Theorem 2.2 gives min qM, Consequently,

Generalized Erdős distance problem.
In vector spaces over finite fields, one may define ∆(x, y) with x − y = (x 1 − y 1 ) 2 + · · · + (x d − y d ) 2 , and one may ask for the smallest possible size of ∆(E, E) in terms of the size of E.
In this context there are additional difficulties to contend with. First, E may be the whole vector space, which would result in the rather small size for the distance set |∆(E, E)| = |E| 1/d . Another consideration is that if q is a prime congruent to 1 (mod 4) , then there exists an i ∈ F q such that i 2 = −1. This allows us to construct a set in F 2 q , Z = {(t, it) : t ∈ F q } and one can easily check that ∆(Z, Z) = {0}. The first non-trivial result on the Erdős distance problem in vector spaces over finite fields was proved by Bourgain, Katz and Tao in [4]. They showed that if |E| q 2−ǫ for some ǫ > 0 and q is a prime ≡ 3 (mod 4). Then In [20] Iosevich and Rudnev gave a distance set result for general fields in arbitrary dimension with explicit exponents. They proved that if |E| ≥ 2q (d+1)/2 , then ∆(x, y) is a strong expander. It may seem reasonable that the exponent (d + 1)/2 is improvable. However, Iosevich, Koh, Rudnev and the first listed author showed in [17] that even for the weaker conclusion that ∆(x, y) is a moderate expander then the exponent (d + 1)/2 is sharp in odd dimensions .
be a symmetric non-degenerate polynomial of degree k and define g(x 1 , x 2 , y 1 , This effectively shows that g is a four-variable moderate expander for |E| k 2 q 3/2 . Here we give a general characterization for which Vu's estimates hold. be a non-degenerate polynomial of degree k and define g(x 1 , x 2 , y 1 , y 2 ) = f (x 1 − y 1 , x 2 − y 2 ). Then the following two propositions are equivalent: (1) f − b does not contain a linear factor for any b ∈ F q . (2) |g(E, F )| min(k −1 q, k −2 |E||F |q −1/2 ) holds for all E, F ⊂ F 2 q .
Proof. (1)⇒(2): Suppose (1) holds true. For any b ∈ F q applying Lemma 2.1 with By Lemma 2.7 and Theorem 2.8 we get which in turn gives (2)⇒(1): Suppose (2) holds true. We are trying to prove (1) also holds true and argue it by contradiction. Suppose there exists b ∈ F q such that f − b contains a linear factor. Thus (f − b) −1 (0) must contain a straight line, say for example L, as a subset. Now we choose two straight lines E, F in F 2 q such that E − F = L. Consequently, g(E, F ) = { b}, a contradiction to (2). We are done.
Combining Theorem 2.6 with the proceding theorem naturally gives the following estimate, which improves the relevant results in [18,40].
Theorem 2.14. Let A be a subset of F q and f ∈ F q [x].
(2) If f contains a simple root not equal to zero then where dB and B d denote the d-fold sum-set and product-set of B respectively.

On the expander x + y 2 in prime fields
From the work of Pudlák ([27]) we know that when |A| ≤ p 1/2 with p prime, one has |{x + y 2 : x, y ∈ A}| |A| 1+ǫ for some ǫ > 0. The arguments relied on the finite field Szemerédi-Trotter incidence theorem established by Bourgain, Katz and Tao ([4]). Therefore the proof did not yield any explicit expansion index. 1 In this section we mainly give for |A| ≤ p 1/2 (for |A| ≥ p 1/2 one may apply Theorem 2.6) that one has the following explicit estimate : Before we proceed to prove the theorem, we recall two results. The first one is a variant of the Balog-Szemerédi-Gowers theorem established by Bourgain and Garaev ([3]), which also played an important role in [14]. The second one is Garaev's type sum-product estimate ( [23]), which was obtained by the second listed author, improving upon the one obtained by Bourgain and Garaev ([3]) and the third listed author ( [29,30]). ).
Then |E| ∼ |A| 2 and Applying Theorem 3.2 with the ambient group F * p , there exists a subset D ⊂ A + A with There are two cases.
In the first case, suppose |D| ≤ p 12/23 , then by Theorem 3.3, we have Combining (4), (5) and notice that D ⊂ A + A, we have Now we apply the Plünnecke-Ruzsa inequality as follows, Therefore combining Theorem 3.1, one has x + y 2 is an expander for all sizes of |A|.
In addition, we notice that if |A| > p 2/3 , then Let us show by adopting the Garaev-Chang example ( [14]) that this is optimal up to the implied constant. Let N < 0.01p be a positive integer, M = [2 √ Np] and let X be the set of x so that x 2 modulo p belongs to the interval [1, M]. Then it is known that |X| M. From the pigeonhole principle, there is a number L such that Remark 3.5. We list some two variable polynomials of degree two: P 1 (x, y) = x+y 2 , P 2 (x, y) = xy + x 2 , P 3 (x, y) = xy + x, P 4 (x, y) = x 2 + y 2 , P 5 (x, y) = x 2 + xy + y 2 . Now we know P 1 is an expander, and from [1,14] we also have P 2 , P 3 are expanders. However P 4 is not an expander. Since if p is large enough, we can first embed B = {x 2 : x ∈ F * p } into the set of natural numbers N, then apply Szemerédi's theorem [38] to find a long arithmetic progression, which in turn implies that P 4 is not an expander. As to P 5 , we currently don't know whether it is an expander or not.