  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 $\mathbb{R}^n$.
