Dimensions of Sets Which Uniformly Avoid Arithmetic Progressions

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 ε Δ 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.