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We show that the set defined by digit restrictions contains arbitrarily longarithmetic progressions if and only if its Assouad dimension is one. Moreover,we show that for any $0\le s\le 1$, there exists some set on $\mathbb{R}$ withHausdorff dimension $s$ whose Fourier dimension is zero and it containsarbitrarily long arithmetic progressions.

On Assouad dimension and arithmetic progressions in sets defined by digit restrictions