Large sets avoiding affine copies of infinite sequences

Abstract

A conjecture of Erdős states that for any infinite set $A\subseteq \mathbb{R}$, there exists $E\subseteq \mathbb{R}$ of positive Lebesgue measure that does not contain any nontrivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A=\{2^{-k}:k\ge 1\}$. In this article, we consider infinite decreasing sequences $A=\{a_k:k\ge 1\}$ in $\mathbb{R}$ that converge to zero at a prescribed rate; namely $\log (a_n/a_{n+1})=e^{\phi (n)}$, where $\phi (n)/n\to 0$ as $n\to \infty$. This condition is satisfied by sequences whose logarithm has polynomial decay, and in particular by the geometric sequence. For any such sequence $A$, we construct a Borel set $\mathcal{O}\subseteq \mathbb{R}$ of Hausdorff dimension 1, but Lebesgue measure zero, that avoids all nontrivial affine copies of $A\cup \{0\}$.

arXiv admin note: substantial text overlap with arXiv:2001.02395