Coarse dimension and definable sets in expansions of the ordered real vector space

Abstract

Suppose $E\subseteq \mathbb{R}$ is nowhere dense. If $(\mathbb{R},<,+,(x\mapsto \lambda x{)}_{\lambda \in \mathbb{R}},E)$ does not define every bounded Borel subset of every ${\mathbb{R}}^n$ then for every $s>0$ we have $$|\{k\in \mathbb{Z},-m\le k\le m-1:[k,k+1]\cap E\ne \varnothing \}|<m^s$$ for sufficiently large $m\in \mathbb{N}$. Then there is an $n\in \mathbb{N}$ and a linear $T:{\mathbb{R}}^n\to \mathbb{R}$ such that $T(E^n)$ is dense. It follows that if $E$ is in addition nowhere dense then $(\mathbb{R},<,+,0,(x\mapsto \lambda x{)}_{\lambda \in \mathbb{R}},E)$ defines every bounded Borel subset of every ${\mathbb{R}}^n$.

arXiv commentNew version, accepted by Illinois J. Math