Bounded distance equivalence of aperiodic Delone sets and bounded remainder sets

Delone sets are generalizations of point lattices: unformly discrete point sets with no large holes. In 1997 Gromov asked whether any Delone set in the Euclidean plane is bilipschitz equivalent to the integer lattice $Z^2$. A simpler but stronger condition than bilipschitz equivalence is bounded distance equivalence. So it is natural to ask which Delone sets in $R^d$ are bounded distance equivalent to (some scaled copy of) $Z^d$. This talk gives a gentle introduction to the problem and presents recent results in this context, mostly for cut-and-project sets on the line. In particular we show a connection between bouded remainder sets and cut-and-project sets that are bounded distance equivalent to some lattice.