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

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 $O(p\mathrm{/}{t}^{\kappa })$. We also find several applications to problems about so–called non–averaging sets, number of collinear triples and mixed energies.