Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions
Classical Analysis and ODEs
2021-03-26 v3 Combinatorics
Metric Geometry
Abstract
We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid -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.
Cite
@article{arxiv.1910.10074,
title = {Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions},
author = {Jonathan M. Fraser and Pablo Shmerkin and Alexia Yavicoli},
journal= {arXiv preprint arXiv:1910.10074},
year = {2021}
}
Comments
14 pages. v3: minor corrections and clarifications