ON THE SUM PRODUCT ESTIMATES AND TWO VARIABLES EXPANDERS

Let Fp be the finite field of a prime order p. Let F : Fp ×Fp → Fp be a function defined by F (x, y) = x(f(x) + by), where b ∈ F∗p and f : Fp → Fp is any function. We prove that if A ⊂ Fp and |A| < p1/2 then |A + A| + |F (A,A)| ' |A| 13 12 . Taking f = 0 and b = 1, we get the well-known sum-product theorem by Bourgain, Katz and Tao, and Bourgain, Glibichuk and Konyagin, and also improve the previous known exponent from 14 13 to 13 12 .


Introduction
The sum product phenomenon has received a great deal of attention, since Erdös and Szemerédi made their well known conjecture that for any ǫ > 0 one has where A is a finite subset of integers,

and
AA = {ab : a ∈ A, b ∈ A}.Later, much work has been done to find the explicit exponents, and the best result to date is due to Solymosi [11], who showed that max(|A + A|, |AA|) |A| 4 3 .In the finite field setting, the problem becomes more complicated and the first non-trivial sum-product estimate was obtained by Bourgain, Katz and Tao [4] with subsequent refinement by Bourgain, Glibichuk and C.-Y. Shen Konyagin [3].They proved that if A ⊂ F p , p prime, and |A| ≤ p 1−δ for some δ > 0, then there exists ǫ = ǫ(δ) > 0 such that max(|A+A|, |AA|) |A| 1+ǫ .Since then there have been several generalizations and applications of this theorem (see [1], [2], [5]- [10], [12]).For example, it was shown by Bourgain [1] that if A, B ⊂ F p and p δ < |B| ≤ |A| < p 1−δ for some δ > 0, then the following bound holds: for some ǫ > 0. In addition, he also showed that the function F (x, y) = x 2 + xy from F p × F p to F p possesses an expanding property in the sense that |F (A, B)| p ǫ for some ǫ > δ whenever |A| ∼ |B| ∼ p δ , 0 < δ < 1.Another generalization was made by Vu [13] who characterized the polynomials which satisfy where k is the degree of the polynomial (see, also [6] for some improvements in the case P (x, y) = xy which corresponds to the sum-product problem).However, this result is nontrivial only when |A| > p 1 2 .In this paper we construct a family of two variables functions of the form , and also prove a stronger sum product estimate in the most nontrivial range |A| < p 12 , where F : F p × F p → F p be a function defined by F (x, y) = x(f (x) + by), where b ∈ F * p and f : F p → F p is any function.Remark 1.1.Taking f = 0 and b = 1, we get the above mentioned sum product theorem from [3] and [4] and also improve the exponent in [9] from 14  13 to 13  12 .In addition, the exponent 13  12 appears in the work of Bourgain and Garaev [2] in the form |A − A| + |AA| |A| 13/12 .Nevertheless, our method is different from the one of [2] and applies equally well to the more general case.

Preliminaries
Throughout this paper A will denote a nonempty subset in the prime field F p .If B is a set then we will denote its cardinality by |B|.Whenever X and Y are quantities we will use X Y, to mean X ≤ CY, where the constant C is universal (i.e.independent of p and A).The constant C may vary from line to line.We will use X Y, to mean X ≤ C(log |A|) α Y, and X ≈ Y to mean X Y and Y X, where C and α may vary from line to line but are universal.
we have that for any In particular such Lemma 2.2.Let X, B 1 , . . ., B k be any subsets of F p .Then there is Lemma 2.3.Let C and D be sets with |D| |C| K and with |C + D| ≤ K|C|.Then there is a C ′ ⊂ C with |C ′ | ≥ 9  10 |C| so that C ′ can be covered by ∼ K 2 translates of D. Similarly, there is a C ′′ ⊂ C with |C ′′ | ≥ 9  10 |C| so that C ′′ can be covered by ∼ K 2 translates of −D.Proof: To prove the first half of the statement, it suffices to show that we can find one translate of D whose intersection with C is at least |C|/K 2 .Once we find such a translate, we remove the intersection and then iterate.We stop when the size of the remaining part of C is less than |C|/10.
To prove the second half of the statement we have to show there is a translate of D whose intersection with −C is at least |C|/K 2 .First, by the Cauchy-Schwartz inequality, we have that The quantity on the left hand side is equal to Thus we can find c ∈ C and d ′ ∈ D so that Proceeding as above, we find c ∈ C and d ∈ D such that and the result follows.for some ǫ > 0 that depends only on δ and on the degree of the polynomial f .

Explicit two variables expanding maps
The key ingredient is the Szemerédi-Trotter incidence theorem in the affine plane F 2 p which was proven in [3], [4].Theorem 3.2.Let P and L be the points and lines in F Proof: We proceed by contradiction.Suppose it is not true.Then we have |{x(f (x) + y) : x, y ∈ A}| |A| 1+ǫ for some small ǫ.Let k be the degree of f and denote C = {x(f (x) + y) : x, y ∈ A}.By the Cauchy-Schwartz inequality, we have Therefore, we can find a 0 ∈ A and A 1 ⊂ A such that Thus, for any Hence, for any x ∈ A we have and the total number of incidences between L and P is at least By applying Theorem 3.2, it follows that if ǫ is too small, it leads a contradiction and this completes the proof.Remark 3.3.In Theorem 3.1 we assume that f is a nonconstant polynomial.If f is a constant, then we mention the recent preprint [7], where explicit bounds have been obtained for this case.  1, where F (x, y) = x(f (x) + by), f is any function from F p to F p , and b ∈ F * p .

Now by applying Lemma 2.2, we get
Applying the same argument as above, we get
is just what we wanted to prove.To prove the second half of the statement we start with the inequality