Large Sets Avoiding Affine Copies of Infinite Sequences

Abstract

A conjecture of Erdös states that for any infinite set $A \subseteq \mathbb R$, there exists a Borel set $E \subseteq \mathbb R$ of positive Lebesgue measure that does not contain any non-trivial affine copy of $A$. The conjecture remains open for most fast-decaying sequences, including the geometric sequence $A = \{2^{-k} : k \geq 1\}$. In this article, we consider infinite decreasing sequences $A = \{a_k: k \geq 1\}$ in $\R$ that converge to zero at a prescribed rate; namely $\log (a_n/a_{n+1}) = e^{\varphi(n)} $, where $\varphi(n)/n\to 0$ 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 Cantor set $K \subseteq \mathbb [0,1]$ with measure arbitrarily close to 1, such that the set of Erdös points $\mathcal{E}\subseteq K$ has Hasudorff dimension 1.