**Definitions:**

A measurable subset $S$ of $\mathbb R$ is said to be *mesoscopic* if there exists a continuous function $f: \mathbb R \to \mathbb R$ such that $f(S)$ is Lebesgue measurable and has nonzero Lebesgue measure.

Question:Is the set of zeroes of a Brownian motion almost surely a mesoscopic set?

*Remark: Note that there exist mesoscopic sets of Lebesgue measure zero - for example the Cantor set with $f$ being the Cantor staircase function.*