Dimensions of Sets Which Uniformly Avoid Arithmetic Progressions
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Publication:5855165
DOI10.1093/IMRN/RNX261zbMath1460.28008arXiv1705.03335OpenAlexW2962741286MaRDI QIDQ5855165
Han Yu, Jonathan M. Fraser, Kota Saito
Publication date: 15 March 2021
Published in: Unnamed Author (Search for Journal in Brave)
Abstract: We provide estimates for the dimensions of sets in $mathbb{R}$ which uniformly avoid finite arithmetic progressions. More precisely, we say $F$ uniformly avoids arithmetic progressions of length $k geq 3$ if there is an $epsilon>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $Delta>0$ inside the $epsilon Delta$ neighbourhood of $F$. Our main result is an explicit upper bound for the Assouad (and thus Hausdorff) dimension of such sets in terms of $k$ and $epsilon$. In the other direction, we provide examples of sets which uniformly avoid arithmetic progressions of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence we obtain a discretised version of a `reverse Kakeya problem': we show that if the dimension of a set in $mathbb{R}^d$ is sufficiently large, then it closely approximates arithmetic progressions in every direction.
Full work available at URL: https://arxiv.org/abs/1705.03335
Fractals (28A80) Arithmetic progressions (11B25) Relations between ergodic theory and number theory (37A44)
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Related Items (6)
Construction of a one-dimensional set which asymptotically and omnidirectionally contains arithmetic progressions ⋮ Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions ⋮ Large sets avoiding Infinite arithmetic / geometric progressions ⋮ Almost Arithmetic Progressions in the Primes and Other Large Sets ⋮ On Assouad dimension and arithmetic progressions in sets defined by digit restrictions ⋮ Dimension growth for iterated sumsets
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