A cut-and-project set is obtained by selecting points of a higher-dimensional point lattice according to
a bounded window in an internal space and projecting the selected points
to a physical space. More precisely, let 
be a point lattice in 
, with coordinate projections 
and 
. Assume that 
is injective on 
and 
is dense. For a window
that is a compact set equal to the closure
of its nonempty interior, the associated model set is
 |
It is a Delone set. The model set is regular when the window boundary has measure zero.
Haynes (2026) constructed a regular planar example from a totally irrational plane in 
with a one-dimensional window that is a compact set
with countable boundary and no internal projection
of a lattice point on that boundary. It has uniform
asymptotic density but is not bi-Lipschitz equivalent
to any planar point lattice. Thus regularity alone
does not guarantee such an equivalence. Haynes (2026) reports AI assistance in the
research and states that he checked the final proofs.
