# Dimensions of sets which uniformly avoid arithmetic progressions

## TL;DR

This paper investigates the dimensions of sets in real space that avoid containing arithmetic progressions, providing bounds on their size and examples of large-dimensional sets that still avoid such progressions, with extensions to higher dimensions.

## Contribution

It offers explicit bounds on the dimensions of sets avoiding arithmetic progressions and constructs examples of large-dimensional such sets, extending the analysis to higher dimensions and related geometric problems.

## Key findings

- Upper bounds for Assouad and Hausdorff dimensions of sets avoiding progressions
- Construction of large-dimensional sets that avoid progressions
- Extension to higher-dimensional analogues and reverse Kakeya problem

## Abstract

We provide estimates for the dimensions of sets in $\mathbb{R}$ which uniformly avoid finite arithmetic progressions. More precisely, we say $F$ uniformly avoids arithmetic progressions of length $k \geq 3$ if there is an $\epsilon>0$ such that one cannot find an arithmetic progression of length $k$ and gap length $\Delta>0$ inside the $\epsilon \Delta$ 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 $\epsilon$. In the other direction, we provide examples of sets which uniformly avoid arithmetic progressions of a given length but still have relatively large Hausdorff dimension. We also consider higher dimensional analogues of these problems, where arithmetic progressions are replaced with arithmetic patches lying in a hyperplane. As a consequence we obtain a discretised version of a `reverse Kakeya problem': we show that if the dimension of a set in $\mathbb{R}^d$ is sufficiently large, then it closely approximates arithmetic progressions in every direction.