A Delone set is a subset 
of Euclidean space 
that is both relatively
dense and, with Euclidean distance, a uniformly discrete metric space. Thus
there are constants 
such that distinct points
of 
are at least 
apart, while every point of 
is within distance 
of a point of 
(Moody 2000).
The first condition prevents arbitrarily close pairs of points; the second prevents arbitrarily large empty balls. An 
-dimensional point
lattice in 
is a Delone set. The definition does not require translation
symmetry, making these sets useful in the study of quasicrystals.
Being a Delone set does not ensure bi-Lipschitz equivalence to a point lattice. Haynes (2026) constructed a regular planar cut-and-project set with uniform asymptotic density and this failure of equivalence.
