Dimensions of sets which uniformly avoid arithmetic progressions
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.
Citation
Fraser , J M , Saito , K & Yu , H 2017 , ' Dimensions of sets which uniformly avoid arithmetic progressions ' , International Mathematics Research Notices , vol. 2017 . https://doi.org/10.1093/imrn/rnx261
Publication
International Mathematics Research Notices
Status
Peer reviewed
ISSN
1073-7928Type
Journal article
Rights
© 2017, the Author(s). This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1093/imrn/rnx261
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