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 $O(p/t^\kappa)$. We also find several applications to problems about so--called non--averaging sets, number of collinear triples and mixed energies. 
Ilya D. Shkredov
math.NT math.CO