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

Ilya D. Shkredov

Published 2016-09-21Version 1

We prove that for an arbitrary $\kappa < \frac{5}{31}$ 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.