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arXiv · 1710.07144

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

Jinjun Li·Min Wu·Ying Xiong
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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.

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Jinjun Li, Min Wu, Ying Xiong. 2017-10-19. On Assouad dimension and arithmetic progressions in sets defined by digit restrictions. https://arxiv.org/abs/1710.07144

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