Fractals and the monadic second order theory of one successor

Abstract

Let $X \subseteq \mathbb{R}^n$ be closed and nonempty. If the $C^k$-smooth points of $X$ are not dense in $X$ for some $k \geq 0$, then $(\mathbb{R},<,+,0,X)$ interprets the monadic second order theory of $(\mathbb{N},+1)$. The same conclusion holds if the Hausdorff dimension of $X$ is strictly greater than the topological dimension of $X$ and $X$ has no affine points. Thus, if $X$ is virtually any fractal subset of $\mathbb{R}^n$, then $(\mathbb{R},<,+,0,X)$ interprets the monadic second order theory of $(\mathbb{N},+1)$. This result is sharp as the standard model of the monadic second order theory of $(\mathbb{N},+1)$ is known to interpret expansions of $(\mathbb{R},<,+,0)$ which define various classical fractals.Comment: Added a proof of definibility of $C^k$-smooth points and improved the introductio