2 For any X1 , X2 , X3 ⊆ Fq , we have |{x1 + x2 x3 : x1 ∈ X1 , x2 ∈ X2 , x3 ∈ X3 }| q, |X1 ||X2 ||X3 | q Proof of Theorem 1.2 We first recall a sum-product estimate of Garaev [12] (see also [35, 39] for different proofs of this result) Lemma 4.1 (see [12]) For arbitrary sets A, B, C ⊂ Fq , we have |A · B||A + C| q|A|, |A|2 |B||C| q Using Fourier analysis techniques, Hart, Li, and Shen [26] obtained the following generalized sum-product-type estimates Lemma 4.2 (see [26, Theorem 2.6]) Let p be the characteristic of Fq and f, g ∈ Fq [x] For any subsets A, B, C ⊂ Fq we have the following (1) If ≤ deg(f ) < deg(g) < p then |f (A) + B||g(A) + C| min(|A|q, |A|2 |B||C|q −1 ) Particularly, one has |f (A) + g(A)| min(|A|1/2 q 1/2 , |A|2 q −1/2 ) (2) 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 Suppose deg(f ) + deg(g) < p Then |f (A) · B||g(A) · C| min(|A|q, |A|2 |B||C|q −1 ) Particularly, one has |f (A) · g(A)| min(|A|1/2 q 1/2 , |A|2 q −1/2 ) We are now ready to give a proof of Theorem 1.2 Let T = {x1 x2 + (x3 − x4 )2 : x1 , x2 , x3 , x4 ∈ A}, X1 = {aa : a, a ∈ A}, X1 = {(a − a )2 : a, a ∈ A}, X2 = X3 = A Then |X1 | ≥ |A − A|/2 From Lemma 4.1, we have (4.1) |X1 ||X1 | |A · A||A − A| q|A|, |A|4 q It follows from Theorem 3.1 that (4.2) |T | = |{x1 + (x2 − x3 )2 : x1 ∈ X1 , x2 ∈ X2 , x3 ∈ X3 }| q, |X1 ||X2 ||X3 | q Copyright © by SIAM Unauthorized reproduction of this article is prohibited 2046 LE ANH VINH Downloaded 06/17/14 to 129.97.58.73 Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php It follows from Theorem 3.2 that (4.3) |T | = |{x1 + x2 x3 : x1 ∈ X1 , x2 ∈ X2 , x3 ∈ X3 }| q, |X1 ||X2 ||X3 | q Putting (4.1), (4.2), and (4.3) together, we have |T | q, |A|4 q 3/2 This proves the first part of Theorem 1.2 Similarly, parts (2) and (3) of Theorem 1.2 follow from parts (1) and (2) of Lemma 4.2 Proof of Theorem 1.5 The proof proceeds by induction on l If |A| q l/(2l−1) then the theorem follows from Theorem 1.3 and Theorem 1.4 Therefore, we can assume that q 2l+1 4l l |A| q 2l−1 For the base case l = 2, then q 5/8 |A| q 2/3 Let X = {(a − b)2 : a, b ∈ A}, Y = Z = A It follows from Theorem 3.1 that (5.1) |Δ(A2 )| = x + (y − z)2 : x ∈ X, y ∈ Y, z ∈ Z q, |X||Y ||Z| q Let X = {ab : a, b ∈ A}, Y = Z = A It follows from Theorem 3.2 that (5.2) |Π(A2 )| = |{x + yz : x ∈ X , y ∈ Y, z ∈ Z}| q, |X ||Y ||Z| q From (4.1), we have (5.3) |X1 ||X1 | |A · A||A − A| q|A|, |A|4 q |A|4 q Putting (5.1), (5.2), and (5.3) together, we have max{|Δ(A2 )|, |Π(A2 )|} q, |A|4 q 3/2 Suppose the statement holds for l ≥ We show that it also holds for l + By the induction hypothesis, we have (5.4) max{|Δ(Al )|, |Π(Al )|} q, |A|2l q (2l−1)/2 From (5.4), Theorems 3.1, and 3.2, we have max{|Δ(Al+1 )|, |Π(Al+1 )|} q, max |Δ(Al )||Y ||Z| |Π(Al )||Y ||Z| , q q q, |A|2 max{|Δ(Al )|, |Π(Al )|} q q, |A|2l+2 q (2l+1)/2 , concluding the proof of the theorem Copyright © by SIAM Unauthorized reproduction of this article is prohibited ON FOUR-VARIABLE EXPANDERS IN FINITE FIELDS 2047 Downloaded 06/17/14 to 129.97.58.73 Redistribution subject to SIAM license or copyright; 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I D Shkredov, On monochromatic solutions of some nonlinear equations in Z/pZ, Math Notes, 88 (2010), pp 603–611 [34] J Solymosi, On the numbers of sums and products, Bull London Math Soc., 37... J Bourgain, Mordell’s exponential sum estimate revisited, J Amer Math Soc., 18 (2005), pp 477–499 [3] J Bourgain, More on the sum-product phenomenon in prime fields and its application, Int J Number... and N Katz, On the Erd˝ os Distinct Distances Problem in the Plane, preprint, arXiv:1011.4105, 2010 ´ rko ¨ zy, Equations in finite fields with restricted solution sets, I (Char[18] K Gyarmati and