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

Abstract

Let $E\subseteq \mathbb{R}$. Suppose there is an $s>0$ such that $$\left|\right\{k\in \mathbb{Z},-m\le k\le m-1:[k,k+1]\cap E\ne \varnothing \left\}\right|\ge m^s$$ for all 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 $T\left(E^n\right)$ is dense. As a corollary, we show 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 ${\mathbb{R}}^n$.