2000 character limit reached
Improved bounds on the dimensions of sets that avoid approximate arithmetic progressions (1910.10074v3)
Published 22 Oct 2019 in math.CA, math.CO, and math.MG
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.