A tame Cantor set

Abstract

A Cantor set is a non-empty, compact subset of $\mathbb{R}$ that has neither interior nor isolated points. In this paper a Cantor set $K\subseteq \mathbb{R}$ is constructed such that every set definable in $(\mathbb{R},<,+,\cdot ,K)$ is Borel. In addition, we prove quantifier-elimination and completeness results for $(\mathbb{R},<,+,\cdot ,K)$, making the set $K$ the first example of a modeltheoretically tame Cantor set. This answers questions raised by Friedman, Kurdyka, Miller and Speissegger. The work in this paper depends crucially on results about automata on infinite words, in particular Büchi's celebrated theorem on the monadic second-order theory of one successor and McNaughton's theorem on Muller automata, which have never been used in the setting of expansions of the real field.