On a Zero-Infinity Law of Olsen

Abstract

Let $\mu$ be a translation-invariant metric measure on $\mathbb{R}$ with the following scaling property: for every $\lambda \in (0,1)$ there exists $b(\lambda)>\lambda$ with $\mu(\lambda X)\geq b(\lambda) \mu(X)$ for all $X \subseteq \mathbb{R}$. If $X$ is a $\mathbb{Z}$-invariant subset of $\mathbb{R}$ with $X/q \subseteq X$ for some $q\in \mathbb{N}\setminus \{1\}$, then $\mu(X)=0$ or $\mu(X\cap O)=\infty$ for every non-empty open set $O$. This refines an earlier result by Olsen.