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 $$ | \{ k \in \mathbb{Z}, -m \leq k \leq m - 1 : [k,k+1] \cap E \neq \emptyset \} | < m^s $$ 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 $\mathbb{R}^n$.