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

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 $$\mathrm{\mid }\{k\in \mathbb{Z},-m\le k\le m-1:[k,k+1]\cap E\mathrm{\ne }\mathrm{\varnothing }\}\mathrm{\mid }<m^s$$ for sufficiently large $m\in \mathbb{N}$. Then there is an $n \in $\mathbb{N}$$andalinear$T : $\mathbb{R}$^n \to $\mathbb{R}$$suchthat$T(E^n)$isdense\mathrm{.}Itfollowsthatif$E$isinadditionnowheredensethen$($\mathbb{R}$,<,+,0,(x \mapsto \lambda x){\lambda \in $\mathbb{R}$}, E)$ defines every bounded Borel subset of every ${\mathbb{R}}^n$.