Three-term arithmetic progressions in subsets of F_q^∞ of large Fourier dimension

Abstract

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