## Illinois Journal of Mathematics

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

Erik Walsberg

#### Abstract

Let $E\subseteq \mathbb{R}$. Suppose there is an $s\gt 0$ such that $\begin{equation*}\bigl\vert \bigl\{k\in \mathbb{Z},-m\leq k\leq m-1:[k,k+1]\cap E\neq\emptyset \bigr\}\bigr\vert \geq m^{s}\end{equation*}$ 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 $T(E^{n})$ is dense. As a corollary, we show that if $E$ is in addition nowhere dense, then $(\mathbb{R},\lt ,+,0,(x\mapsto \lambda x)_{\lambda \in \mathbb{R}},E)$ defines every bounded Borel subset of every $\mathbb{R}^{n}$.

#### Article information

Source
Illinois J. Math., Volume 64, Number 2 (2020), 141-149.

Dates
Revised: 12 November 2019
First available in Project Euclid: 1 May 2020

https://projecteuclid.org/euclid.ijm/1588298624

Digital Object Identifier
doi:10.1215/00192082-8303453

Mathematical Reviews number (MathSciNet)
MR4092952

Zentralblatt MATH identifier
07210953

Subjects
Primary: 03C64: Model theory of ordered structures; o-minimality

#### Citation

Walsberg, Erik. Coarse dimension and definable sets in expansions of the ordered real vector space. Illinois J. Math. 64 (2020), no. 2, 141--149. doi:10.1215/00192082-8303453. https://projecteuclid.org/euclid.ijm/1588298624

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