Related papers: Fast repetitivity in non-rectifiable Delone sets

In this short paper, we consider the functional density on sets of uniformly bounded triangulations with fixed sets of vertices. We prove that if a functional attains its minimum on the Delaunay triangulation, for every finite set in the…

We establish that any subset of $\mathbb{R}^d$ of positive upper Banach density necessarily contains an isometric copy of all sufficiently large dilates of any fixed two-dimensional rectangle provided $d\geq4$. We further present an…

We prove algorithmic and hardness results for the problem of finding the largest set of a fixed diameter in the Euclidean space. In particular, we prove that if $A^*$ is the largest subset of diameter $r$ of $n$ points in the Euclidean…

The tree reconstruction problem is to find an embedded straight-line tree that approximates a given cloud of unorganized points in $\mathbb{R}^m$ up to a certain error. A practical solution to this problem will accelerate a discovery of new…

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of the set of all bounded 1-Lipschitz functions $f\colon X \to…

Given two arbitrary closed sets in Euclidean space, a simple transversality condition guarantees that the method of alternating projections converges locally, at linear rate, to a point in the intersection. Exact projection onto nonconvex…

We consider the set of monic irreducible polynomials $P$ over a finite field $\mathbb{F}_q$ such that the multiplicative order modulo $P$ of some a in $\mathbb{F}_q(T)$ is divisible by a fixed positive integer $d$. Call $R_q(a,d)$ this set.…

The densest local packings of N identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of…

Two Delone sets are bounded distance equivalent to each other if there is a bijection between them such that the distance of corresponding points is uniformly bounded. Bounded distance equivalence is an equivalence relation. We show that…

The study is motivated by the known fact that, in the noncompact case, the main minimum-problem of the theory of interior capacities of condensers in a locally compact space is in general unsolvable, and this occurs even under very natural…

Fix integers $b>a\geq1$ with $g:=\gcd(a,b)$. A set $S\subseteq\mathbb{N}$ is \emph{$\{a,b\}$-multiplicative} if $ax\neq by$ for all $x,y\in S$. For all $n$, we determine an $\{a,b\}$-multiplicative set with maximum cardinality in $[n]$, and…

With a view to establishing measure theoretic approximation properties of Delone sets, we study a setup which arises naturally in the problem of averaging almost periodic functions along exponential sequences. In this setting, we establish…

Maximum likelihood iteration is one of the most commonly used reconstruction algorithms in quantum tomography. The main appeal of the method is that it is easy to implement and that it converges reliably to a physically meaningful density…

We provide, for any $r\in (0,1)$, lower and upper bounds on the maximal density of a packing in the Euclidean plane of discs of radius $1$ and $r$. The lower bounds are mostly folk, but the upper bounds improve the best previously known…

Delone sets are locally finite point sets, such that (a) any two points are separated by a given minimum distance, and (b) there is a given radius so that every ball of that radius contains at least one point. Important examples include the…

We apply the topology of convergence on compact sets to define unpredictable functions [5, 6]. The topology is metrizable and easy for applications with integral operators. To demonstrate the effectiveness of the approach, the existence and…

Given any 1-random set $X$ and any $r\in(0,1)$, we construct a set of intrinsic density $r$ which is computable from $r\oplus X$. For almost all $r$, this set will be the first known example of an intrinsic density $r$ set which cannot…

Delone sets are discrete point sets $X$ in $\mathbb{R}^d$ characterized by parameters $(r,R)$, where (usually) $2r$ is the smallest inter-point distance of $X$, and $R$ is the radius of a largest ``empty ball" that can be inserted into the…

We use dense Sidon sets to construct small weighted projective 2-designs. This represents quantitative progress on Zauner's conjecture.

We prove that certain families of homogenous affine iterated function systems in $\mathbb{R}^d$ have the property that the open set condition and the existence of exact overlaps both occur densely in the space of translation parameters.…