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

## TL;DR

This paper investigates the properties of definable sets in expansions of the ordered real vector space, establishing conditions under which such expansions can or cannot define all bounded Borel sets, and relating these to coarse dimension and density.

## Contribution

It introduces a new criterion linking coarse dimension, nowhere dense sets, and definability of bounded Borel sets in expansions of the real vector space.

## Key findings

- If certain conditions hold, the set's intersection with integer intervals grows slower than any polynomial.
- Existence of a linear map making the set dense in the real line.
- If the set is nowhere dense, the expansion defines all bounded Borel subsets.

## 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 $\mathbb{R}^n$.