Related papers: Constructing bounded remainder sets and cut-and-pr…
Recent results of several authors have led to constructions of parallelotopes which are bounded remainder sets for totally irrational toral rotations. In this brief note we explain, in retrospect, how some of these results can easily be…
We prove that in any totally irrational cut-and-project setup with codimension (internal space dimension) one, it is possible to choose sections (windows) in non-trivial ways so that the resulting sets are bounded displacement to lattices.…
In this paper we give an explicit construction of bounded remainder sets of all possible volumes, for any irrational rotation on the adelic torus $\mathbb A/\mathbb Q$. Our construction involves ideas from dynamical systems and harmonic…
In this paper we give a simple, explicit construction of polytopal bounded remainder sets of all possible volumes, for any irrational rotation on the $d$ dimensional adelic torus $\mathbb{A}^d/\mathbb{Q}^d$. Our construction involves ideas…
By using a combination of algebraic, geometric, and dynamical techniques, together with input from higher dimensional Diophantine approximation, we give a complete characterization of all linearly repetitive cut and project sets with…
This paper uses a connection between bounded remainder sets in $\mathbb{R}^d$ and cut-and-project sets in $\mathbb{R}$ together with the fact that each one-dimensional Pisot substitution sequence is bounded distance equivalent to some…
We show that real tight frames that generate lattices must be rational, and use this observation to describe a construction of lattices from vertex transitive graphs. In the case of irreducible group frames, we show that the corresponding…
We show that the total Cartier index of varieties with rational singularities in a bounded family is bounded. This solves a problem of Han and Jiang. The overall structure of the proof, which treats the surface case and the…
Given an initial family of sets, we may take unions, intersections and complements of the sets contained in this family in order to form a new collection of sets; our construction process is done recursively until we obtain the last family.…
We study deterministic constructions of graphs for which the unique completion of low rank matrices is generically possible regardless of the values of the entries. We relate the completability to the presence of some patterns (particular…
A new construction of naturally reductive spaces is presented. This construction gives a large amount of new families of naturally reductive spaces. First the infinitesimal models of the new naturally reductive spaces are constructed. A…
The aim of this article is to show the existence, and also give an explicit construction, of infinite sets of orthogonal exponentials for certain families of convex polytopes which include simple-rational polytopes and also non simple…
We construct geometric realizations for the infimax family of substitutions by generalizing the Rauzy-Canterini-Siegel method for a single substitution to the S-adic case. The composition of each countably infinite subcollection of…
We study bounded remainder sets with respect to an irrational rotation of the $d$-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one. First we…
This paper gives a complete classification of linear repetitivity (LR) for a natural class of aperiodic Euclidean cut and project schemes with convex polytopal windows. Our results cover those cut and project schemes for which the lattice…
Consider the d-dimensional lattice Z^d where each vertex is ``open'' or ``closed'' with probability p or 1-p, respectively. An open vertex v is connected by an edge to the closest open vertex w such that the dth co-ordinates of v and w…
In a projective plane PG(2,K) defined over an algebraically closed field K of characteristic 0, we give a complete classification of 3-nets realizing a finite group. An infinite family, due to Yuzvinsky, arises from plane cubics and…
We define the bounded coarse structure attached to a family of pseudometrics and give some counterexamples to conjectures that arise naturally.
We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\{x_1+t\}, \{x_2+t\alpha \})_{t \geq 0}$ in the unit square. In particular, we show that for almost all $\alpha$ and every starting point $(x_1,…
We give an upper bound for the degree of rational curves in a family that covers a given birational ruled surface in projective space. The upper bound is stated in terms of the degree, sectional genus and arithmetic genus of the surface. We…