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Dense forests and Danzer sets
, 2014
"... Abstract. 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(T d). We prove that natural candidates, such as discrete sets that arise from substitutions and from cutandproject constructions, are ..."
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Abstract. 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(T d). We prove that natural candidates, such as discrete sets that arise from substitutions and from cutandproject 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 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 wellknown combinatorial problem, and deduce the existence of Danzer sets with growth rate O(T d log T), improving the previous bound of O(T d logd−1 T). 1.
doi:10.1093/imrn/rnu140 Visibility and Directions in Quasicrystals
"... It is well known that a positive proportion of all points in a ddimensional lattice is visible from the origin, and that these visible lattice points have constant density in Rd. In the present paper, we prove an analogous result for a large class of quasicrystals, including the vertex set of a Pen ..."
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It is well known that a positive proportion of all points in a ddimensional lattice is visible from the origin, and that these visible lattice points have constant density in Rd. In the present paper, we prove an analogous result for a large class of quasicrystals, including the vertex set of a Penrose tiling. We furthermore establish that the statistical properties of the directions of visible points are described by certain SL(d,R)invariant point processes. Our results imply in particular existence and continuity of the gap distribution for directions in certain 2D cutandproject sets. This answers some of the questions raised by Baake et al. [1]. 1
POINTWISE EQUIDISTRIBUTION WITH AN ERROR RATE AND WITH RESPECT TO UNBOUNDED FUNCTIONS
"... Abstract. Consider G = SL d (R) and Γ = SL d (Z). It was recently shown by the secondnamed author ..."
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Abstract. Consider G = SL d (R) and Γ = SL d (Z). It was recently shown by the secondnamed author