Arithmetic patches, weak tangents, and dimension

TL;DR

This paper explores the connections between arithmetic progressions, tangent sets, and Assouad dimension, establishing conditions under which sets contain large arithmetic patches or have certain geometric properties, with applications to integer sets.

Contribution

It extends existing results by linking Assouad dimension to the presence of arithmetic patches and tangent set structure, providing new insights into geometric and combinatorial properties.

Findings

Sets with Assouad dimension less than the ambient dimension lack large arithmetic patches.

Having full Assouad dimension is equivalent to tangent sets with interior and containing large arithmetic patches.

Applications include a weak solution to the Erdős-Turán conjecture on arithmetic progressions.

Abstract

We investigate the relationships between several classical notions in arithmetic combinatorics and geometry including: the presence (or lack of) arithmetic progressions (or patches in dimensions $\geq 2$); the structure of tangent sets; and the Assouad dimension. We begin by extending a recent result of Dyatlov and Zahl by showing that a set cannot contain arbitrarily large arithmetic progressions (patches) if it has Assouad dimension strictly smaller than the ambient spatial dimension. Seeking a partial converse, we go on to prove that having Assouad dimension equal to the ambient spatial dimension is equivalent to having weak tangents with non-empty interior and to `asymptotically' containing arbitrarily large arithmetic patches. We present some applications of our results concerning sets of integers, which include a weak solution to the Erd\"os-Tur\'an conjecture on arithmetic progressions.