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

Abstract

We show that subsets of \(\mathbf{F}_q^{\infty}\) 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.