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.