Improved Bounds on the Dimensions of Sets that Avoid Approximate Arithmetic Progressions

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 Szemerédi bounds. In particular, we answer a question of Fraser, Saito and Yu (IMRN 14:4419–4430, 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.