Arithmetic progressions in multiplicative groups of finite fields

Abstract

Let $G$ be a multiplicative subgroup of the prime field ${\mathbb{F}}_p$ of size $|G|>p^{1-\kappa }$ and $r$ an arbitrarily fixed positive integer. Assuming $\kappa =\kappa (r)>0$ and $p$ large enough, it is shown that any proportional subset $A\subset G$ contains non-trivial arithmetic progressions of length $r$. The main ingredient is the Szemerédi-Green-Tao theorem.