Sums and products in finite fields: an integral geometric viewpoint

Abstract

We prove that if A ⊂ F q is such that

On the Solvability of Bilinear Equations in Finite Fields

“…On the other hand, we show that our approach also gives an alternative proof of the corresponding results of [9] and [18] for available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0017089508004382…”

“…which leads to (9). We remark that in a recent work of Garaev [5] considering equation (6) (for some special sets A, B, C, D) has been the main tool in obtaining new results on the sumproduct problem in finite fields (see also [2-4, 9, 11, 15, 16]).…”

“…Using exponential sums, Hart and Iosevich [9] have shown that for any 2n sets A i , B i ⊆ ‫ކ‬ q , i = 1, . .…”

“…has a solution for any λ ∈ ‫ކ‬ * q (although the proof is given only in the case of A 1 = B 1 = · · · = A n = B n , the method and results immediately extend to the general case, see [9,Remark 1.3]). These results have been put in a more general context in a recent work of Hart et al [10].…”

Untitled

“…Bourgain [2005] showed that for A ⊆ ‫ކ‬ q such that |A| q 3/4 , one has A · A + A · A + A · A = ‫ކ‬ q ; in particular, if |A| ≈ q 3/4 , then |A · A + A · A + A · A| |A| 4/3 . In [Hart and Iosevich 2008] it was shown that if |A| q 3/4 , then A · A + A · A = ‫ކ‬ * q ; in particular, if |A| ≈ q 3/4 , then |A · A + A · A| |A| 4/3 . Due to the misbehavior of the zero element, it is not possible to guarantee that A · A + A · A = ‫ކ‬ q unless A is a positive proportion of the elements of ‫ކ‬ q .…”