Arithmetic progressions in multiplicative groups of finite fields

TL;DR

This paper proves that large subsets of multiplicative groups in finite fields contain non-trivial arithmetic progressions of any fixed length, extending the understanding of additive structures in multiplicative groups.

Contribution

It demonstrates that proportional subsets of large multiplicative groups in finite fields necessarily contain arithmetic progressions of any fixed length, using the Szemerédi-Green-Tao theorem.

Findings

Large subsets of multiplicative groups contain long arithmetic progressions

The result holds for sufficiently large primes p

The proof relies on the Szemerédi-Green-Tao theorem

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 Szemer\'{e}di-Green-Tao theorem.