An application of the sum-product phenomenon to sets having no solutions of several linear equations

Abstract

We prove that for an arbitrary κ 1 3 any subset of F p avoiding t linear equations with three variables has size less than O(p/t κ ). We also find several applications to problems about so-called non-averaging sets, number of collinear triples and mixed energies.

An energy decomposition theorem for matrices and related questions

“…It follows from this theorem that for any set A of matrices in M 2 (F q ), we always can find a subset with either small additive energy or small multiplicative energy. In the setting of finite fields, such a result has many applications in studying exponential sums and other topics, for instance, see [4,7,9,12,13,14,15] and references therein. By the Cauchy-Schwarz inequality, we have the following direct consequence on a sum-product estimate, namely, for A ⊆ GL 2 (F q ), we have…”

“…Now, (13) follows from either of (17) or (18). To prove (12), we first note that in either of the cases above we have |X * | ≫ κ(log |X|) −1/2 . Then using the lower bound on κ, (16) Lemma 4.4.…”