# The images of definable sets in the torus, and their associated Hausdorff limits

For a lattice $\Lambda\subseteq {\mathbb R}^n$, let $\pi: {\mathbb R}^n \to T = {\mathbb R}^n/\Lambda$ be the projection onto the corresponding torus. In a previous work we considered the closure of $\pi(X)$ in $T$, when $X$ is definable in an o minimal structure, and described it in terms of affine real sets, associated to the types on $X$. The description was uniform in $\Lambda$.

We now consider a definable family of sets $\{X_t: t \in (0,\infty)\}$ in ${\mathbb R}^n$, and aim to describe all possible Huasdorff limits in $T$ of $\pi(X_t)$, as $t$ tends to $\infty$. We do that (again uniformly in $\Lambda$) in terms of affine sets in elementary extensions, associated to types on $X_t$, for $t>>0$. Curiously, we conclude that any two such limit sets are, up tp a finite partition, translates of each another.

The problem is a topological analogue, suggested by Amos Nevo, to measure theoretic problems in Ergodic theory, regarding so-called dilations on tori and nilmanifolds.

Joint with Sergei Starchenko.