Paper detail

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

We prove that for an arbitrary $κ\le \frac{1}{3}$ any subset of $\mathbf{F}_p$ 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.