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.