Almost Erdős sets

Abstract A set $S \subseteq \mathbb{R}$ is almost Erdős if, for every $\varepsilon \gt 0$ , there exists a set $E \subseteq \mathbb{R}$ of positive Lebesgue measure such that $\{x \in S : ax+b \notin E\}$ is nonempty for all $|a| \gt \varepsilon$ and $b \in \mathbb{R}$ . In this note, we show that any decreasing null sequence $(x_n)$ with decay rate greater than $1/2$ is an almost Erdős set.