Construction of a one-dimensional set which asymptotically and omnidirectionally contains arithmetic progressions

TL;DR

This paper constructs a subset of Euclidean space with Assouad dimension 1 that asymptotically and omnidirectionally contains all possible arithmetic progressions and patches, establishing the minimal dimension for such sets.

Contribution

It provides the first explicit construction of a set with Assouad dimension 1 that contains all directions of arithmetic progressions and patches asymptotically.

Findings

Constructed a set with Assouad dimension 1 containing all directions of arithmetic progressions.

Proved that any set containing all directions of progressions must have Assouad dimension at least 1.

Extended results to higher-dimensional arithmetic patches.

Abstract

In this paper, we construct a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions but has Assouad dimension 1. More precisely, we say that $F$ asymptotically and omnidirectionally contains arithmetic progressions if we can find an arithmetic progression of length $k$ and gap length $\Delta>0$ with direction $e\in S^{d-1}$ inside the $\epsilon \Delta$ neighbourhood of $F$ for all $\epsilon>0$, $k\geq 3$ and $e\in S^{d-1}$. Moreover, the dimension of our constructed example is the lowest-possible because we prove that a subset of $\mathbb{R}^d$ which asymptotically and omnidirectionally contains arithmetic progressions must have Assouad dimension greater than or equal to 1. We also get the same results for arithmetic patches, which are the higher dimensional extension of arithmetic progressions.