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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 ℝ^{d} is sufficiently large, then it closely approximates APs in every direction.
Original language  English 

Number of pages  12 
Journal  International Mathematics Research Notices 
Volume  2017 
Early online date  2 Nov 2017 
DOIs  
Publication status  Epub ahead of print  2 Nov 2017 
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Dive into the research topics of 'Dimensions of sets which uniformly avoid arithmetic progressions'. Together they form a unique fingerprint.Projects
 1 Finished

Fractal Geometry and Dimension: Fractal Geometry and dimension theory
1/09/16 → 30/06/18
Project: Fellowship
Profiles

Jonathan Fraser
 School of Mathematics and Statistics  Director of Research
 Pure Mathematics  Professor
 Centre for Interdisciplinary Research in Computational Algebra
Person: Academic