Three-term arithmetic progressions in subsets of ${\mathbb{F}}_q^{\infty }$ of large Fourier dimension

Abstract

We show that subsets of ${\mathbb{F}}_q^{\infty }$ of large Fourier dimension must contain three-term arithmetic progressions. This contrasts with a construction of Shmerkin of a subset of $\mathbb{R}$ of Fourier dimension $1$ with no three-term arithmetic progressions.

14 Pages