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

Abstract

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

32 pages

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