TY - JOUR

T1 - Dense forests and Danzer sets

AU - Solomon, Yaar

AU - Weiss, Barak

N1 - Publisher Copyright:
© 2016 Société Mathématique de France.

PY - 2016/9/1

Y1 - 2016/9/1

N2 - A set Y ∪ ℝd that intersects every convex set of volume 1 is called a Danzer set. It is not known whether there are Danzer sets in Rd with growth rate O(Td). 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 a uniformly discrete dense forest, 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 O(Td log T), improving the previous bound of O(Tdlogd-1T).

AB - A set Y ∪ ℝd that intersects every convex set of volume 1 is called a Danzer set. It is not known whether there are Danzer sets in Rd with growth rate O(Td). 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 a uniformly discrete dense forest, 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 O(Td log T), improving the previous bound of O(Tdlogd-1T).

UR - http://www.scopus.com/inward/record.url?scp=85020524119&partnerID=8YFLogxK

U2 - 10.24033/asens.2303

DO - 10.24033/asens.2303

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AN - SCOPUS:85020524119

SN - 0012-9593

VL - 49

SP - 1053

EP - 1074

JO - Annales Scientifiques de l'Ecole Normale Superieure

JF - Annales Scientifiques de l'Ecole Normale Superieure

IS - 5

ER -