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 -approximations of arithmetic progressions. Some of these estimates are in terms of Szemer\'edi 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.
0
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.