Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions

TL;DR

This paper establishes improved quantitative bounds on the Hausdorff dimension of sets avoiding approximate arithmetic progressions, connecting these bounds to Szemerédi's theorem and other fractal dimensions.

Contribution

It provides new, sharper bounds on the Hausdorff dimension of such sets, answering a previous open question and relating different notions of fractal dimension.

Findings

Established bounds improve previous estimates.

Connected Hausdorff dimension with box and Assouad dimensions.

Provided a lower bound for Fourier dimension.

Abstract

We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid $\varepsilon$-approximations of arithmetic progressions. Some of these estimates are in terms of Szemer\'{e}di bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN, 2019) and considerably improve their bounds. We also show that Hausdorff dimension is equivalent to box or Assouad dimension for this problem, and obtain a lower bound for Fourier dimension.