Dimensions of sets which uniformly avoid arithmetic progressions

Jonathan MacDonald Fraser, Kota Saito, Han Yu

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)
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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.
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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