1,049 research outputs found
Dense forests and Danzer sets
A set that intersects every convex set of volume
is called a Danzer set. It is not known whether there are Danzer sets in
with growth rate . We prove that natural candidates,
such as discrete sets that arise from substitutions and from cut-and-project
constructions, are not Danzer sets. For cut and project sets our proof relies
on the dynamics of homogeneous flows. We consider a weakening of the Danzer
problem, the existence of uniformly discrete dense forests, and we use
homogeneous dynamics (in particular Ratner's theorems on unipotent flows) to
construct such sets. We also prove an equivalence between the above problem and
a well-known combinatorial problem, and deduce the existence of Danzer sets
with growth rate , improving the previous bound of
A Danzer set for Axis Parallel Boxes
We present concrete constructions of discrete sets in () that intersect every aligned box of volume in , and which have optimal growth rate
- β¦