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Dimensions of sets which uniformly avoid arithmetic progressions

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Jonathan MacDonald Fraser, Kota Saito, Han Yu

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Abstract

We provide estimates for the dimensions of sets in ℝ which uniformly avoid finite arithmetic progressions (APs). More precisely, we say F uniformly avoids APs of length k≥3 if there is an ϵ>0 such that one cannot find an AP of length k and gap length Δ>0 inside the ϵΔ 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 ϵ. In the other direction, we provide examples of sets which uniformly avoid APs of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where APs are replaced with arithmetic patches lying in a hyperplane. As a consequence, we obtain a discretized version of a “reverse Kakeya problem:” we show that if the dimension of a set in ℝd is sufficiently large, then it closely approximates APs in every direction.
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Original languageEnglish
Number of pages12
JournalInternational Mathematics Research Notices
Volume2017
Early online date2 Nov 2017
DOIs
Publication statusE-pub ahead of print - 2 Nov 2017

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