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

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