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On Assouad dimension and arithmetic progressions in sets defined by digit restrictions (1710.07144v2)
Published 19 Oct 2017 in math.CA
Abstract: We show that the set defined by digit restrictions contains arbitrarily long arithmetic 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 $\mathbb{R}$ with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.