Is the zero set of the Conway base 13 function Borel?

Question

Let $Z$ be the zero set and $S$ the set of numbers that can be written using only the digits 0 through 9 (base 13), which has measure zero. Then $Z^c\subset\mathbb{Q}+S$, which has measure zero.

Answer 1

Andres Caicedo pointed out in his answer to the math StackExchange question Examples of Baire class 2 functions that the Conway base $13$ function is a Baire $2$ function. Therefore, the inverse image of any open set is a $G_{\delta \sigma}$ set. From this it easily follows (use De Morgan's laws) that the inverse image of any closed set is an $F_{\sigma \delta}$ set. Since the zero set of a function is the inverse image of the closed set $\{0\},$ it follows that the zero set of the Conway base $13$ function is an $F_{\sigma \delta}$ set. Thus, the zero set is not only a Borel set, but it belongs to a fairly low Borel class.