We review the problem of identifying distance rates for which almost sure asymptotic closeness properties between orbits of measure-preserving dynamical systems can be ensured. More precisely, we consider the set $E_n$ of pairs of points $x,y$ whose orbits up to time $n$ have minimal distance to each other less than the threshold $r_n$. If $(r_n)_n$ shrinks sufficiently slowly, almost every pair $x, y$ will meet this condition for all large enough $n$, i.e., $(\mu\times \mu)(\limsup_n E_n)=1$. On the other hand, if $(r_n)_n$ shrinks too quickly, then the measure of this set is zero. In joint work with Mike Todd, Maxim Kirsebom, and Tomas Persson, we obtain bounds on the sequence $(r_n)_n$ to guarantee that $\limsup_{n}E_n$ and $\liminf_{n} E_n$ are sets of measure 0 or 1.