Arithmetic progressions in sets of fractional dimension

TL;DR

This paper demonstrates that certain fractal sets with Hausdorff dimension close to 1, supporting measures with specific decay properties, necessarily contain non-trivial 3-term arithmetic progressions.

Contribution

It establishes new conditions under which fractal sets of fractional dimension contain arithmetic progressions, extending classical results to more complex sets.

Findings

Sets with Hausdorff dimension near 1 contain 3-term arithmetic progressions

Supports measure conditions imply existence of progressions

Fourier decay properties are crucial for the result

Abstract

Let $E\subset\rr$ be a closed set of Hausdorff dimension $\alpha$. We prove that if $\alpha$ is sufficiently close to 1, and if $E$ supports a probabilistic measure obeying appropriate dimensionality and Fourier decay conditions, then $E$ contains non-trivial 3-term arithmetic progressions.