On Assouad dimension and arithmetic progressions in sets defined by digit restrictions

TL;DR

This paper establishes a precise link between the Assouad dimension and the presence of arbitrarily long arithmetic progressions in digit-restricted sets, and constructs sets with specified Hausdorff dimension containing such progressions despite having zero Fourier dimension.

Contribution

It proves that digit-restricted sets contain arbitrarily long arithmetic progressions if and only if their Assouad dimension is one, and constructs sets with any Hausdorff dimension s that contain progressions but have zero Fourier dimension.

Findings

Digit-restricted sets contain arbitrarily long arithmetic progressions iff Assouad dimension is one.

Existence of sets with any Hausdorff dimension s containing progressions but zero Fourier dimension.

Characterization of arithmetic progressions in fractal sets based on dimensional properties.

Abstract

We show that the set defined by digit restrictions contains arbitrarily long arithmetic progressions if and only if its Assouad dimension is one. Moreover, we show that for any $0\le s\le 1$, there exists some set on $\mathbb{R}$ with Hausdorff dimension $s$ whose Fourier dimension is zero and it contains arbitrarily long arithmetic progressions.