Year: 2026
Type: preprint
Abstract: Let $(a_n)$ be a positive null sequence with $\liminf_{n\to\infty}a_{n+1}/a_n>0$. For every $η>0$, we construct a compact set $E\subset[0,1]$ with Lebesgue measure greater than $1-η$ containing no nontrivial affine copy of $\{a_n:n\ge1\}$. No monotonicity is assumed. The proof adapts a finite-tree construction for geometric sequences to grids chosen from a subsequence with two-sided ratio bounds. We also obtain one such set for any prescribed countable family of sequences, with infinitely many distinct points omitted from every affine copy.
Source: arXiv (Cornell University)
Authors: Jiang, Kan, Xi, Lifeng
Institution: Ningbo University
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Awards: G1097753822, G1533368387, G4442544331
OpenAlex ID: W7221192482