Dense forests and Danzer sets

  title={Dense forests and Danzer sets},
  author={Yaar Solomon and Barak Weiss},
A set $Y\subseteq\mathbb{R}^d$ that intersects every convex set of volume $1$ is called a Danzer set. It is not known whether there are Danzer sets in $\mathbb{R}^d$ with growth rate $O(T^d)$. 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… 
On problems of Danzer and Gowers and dynamics on the space of closed subsets of $\mathbb{R}^d$
Considering the space of closed subsets of $\mathbb{R}^d$, endowed with the Chabauty-Fell topology, and the affine action of $SL_d(\mathbb{R})\ltimes\mathbb{R}^d$, we prove that the only minimal
Danzer's Problem, Effective Constructions of Dense Forests and Digital Sequences
The goal of this paper is to provide constructions of dense and optical forests which yield the best known results in any dimension d ≥ 2 both in terms of visibility and density bounds and effectiveness.
Considering the space of closed subsets of R, endowed with the Chabauty-Fell topology, and the affine action of SLd(R)⋉R , we prove that the only minimal subsystems are the fixed points {∅} and {R}.
On Visibility Problems with an Infinite Discrete, set of Obstacles
A Ramsey type result concerning uniformly separated subsets of $\mathbb{R}^2$ whose growth is faster than linear is presented and a number of other results clarifying how the size of a sets may affect the sets.
Uniformly Discrete Forests with Poor Visibility
A (planar) dense forest is a set F ⊂ R2 so that there exists a function f : (0,1) → R+ such that for any ε ∈ (0,1) and any line segment in the plane of length at least f (ε), there is a point x ∈ F
A Contribution to Metric Diophantine Approximation: the Lebesgue and Hausdorff Theories
  • F. Adiceam
  • Mathematics
    Irish Mathematical Society Bulletin
  • 2016
This thesis is concerned with the theory of Diophantine approximation from the point of view of measure theory. After the prolegomena which conclude with a number of conjectures set to understand
Visibility Properties of Spiral Sets
A spiral in R is defined as a set of the form { d+1 √ n · un}n≥1 , where (un)n≥1 is a spherical sequence. Such point sets have been extensively studied, in particular in the planar case d = 1, as
Cut-and-project quasicrystals, lattices, and dense forests
Dense forests are discrete subsets of Euclidean space which are uniformly close to all sufficiently long line segments. The degree of density of a dense forest is measured by its visibility function.
Classification and statistics of cut and project sets
We define Ratner-Marklof-Strömbergsson measures (following [MS14]). These are probability measures supported on cutand-project sets in R pd ě 2q which are invariant and ergodic for the action of the
Uniformly Discrete Forests with Poor Visibility
  • N. Alon
  • Computer Science, Mathematics
    Combinatorics, Probability and Computing
  • 2017
We prove that there is a set F in the plane so that the distance between any two points of F is at least 1, and for any positive ϵ < 1, and every line segment in the plane of length at least


Ratner's Theorems on Unipotent Flows
The theorems of Berkeley mathematician, Marina Ratner have guided key advances in the understanding of dynamical systems. Unipotent flows are well-behaved dynamical systems, and Ratner has shown that
Lectures on discrete geometry
  • J. Matousek
  • Computer Science, Mathematics
    Graduate texts in mathematics
  • 2002
This book is primarily a textbook introduction to various areas of discrete geometry, in which several key results and methods are explained, in an accessible and concrete manner, in each area.
An Introduction to the Geometry of Numbers
Notation Prologue Chapter I. Lattices 1. Introduction 2. Bases and sublattices 3. Lattices under linear transformation 4. Forms and lattices 5. The polar lattice Chapter II. Reduction 1. Introduction
The Probabilistic Method
A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.
Algebraic groups and number theory
(Chapter Heading): Algebraic Number Theory. Algebraic Groups. Algebraic Groups over Locally Compact Fields. Arithmetic Groups and Reduction Theory. Adeles. Galois Cohomology. Approximation in
Algebraic groups with few subgroups
Every semisimple linear algebraic group over a field F contains nontrivial connected subgroups, namely, maximal tori. In the early 1990s, J. Tits proved that some groups of type E8 have no others. We
A Set Containing Recfiable Arcs QC-locally But Not QC-globally
We construct a Sierpinski carpet E ⊂ R2 of area zero and a K0 > 1 with the property that every K0-quasiconformal image of E contains rectifiable curves, but such that E has some quasiconformal image
Epsilon-nets and simplex range queries
A new technique for half-space and simplex range query using random sampling to build a partition-tree structure and introduces the concept of anε-net for an abstract set of ranges to describe the desired result of this random sampling.
Free Path Lengths in Quasicrystals
Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g., at the points of a spatial Poisson process) or at the
Représentations linéaires irréductibles d'un groupe réductif sur un corps quelconque.
Cet article, issu d'une sorie d'exposes de seminaire faits a FUniversite de Yale au printemps 1967, n'est guere plus qu'une mise au goüt du jour du memoire [5] oü E. Cartan determine les