We consider two number-theoretic problems arising from Fuglede's spectral set conjecture: characterizing finite sets that tile integers, and finding polynomials with (0,1) coefficients whose roots have a certain multiplicative structure. We verify several special cases of the relevant conjectures; in particular, we find necessary and sufficient conditions for a set A to tile the integers if A is a direct sum of cyclic subsets.
When $\mathbb{Z}^d$ is represented as a finite disjoint union of translated integer sublattices, the translated sublattices must possess some special properties. Such a representation is called a \emph{lattice tiling}. We develop a theoretical framework, based on multiple residues and dual groups, to provide a set of necessary and sufficient conditions for such a lattice tiling to exist. We also investigate the question of when a lattice tiling must possess at least two translated sublattices which are translates of one another.
Suppose that A is a finite set of integers of diameter D. Suppose also that the set of integers B is such that A+B is a tiling of the integers, that is each integer is uniquely expressible as a+b, with a in A, b in B. It is well known that B must be a periodic set in this case. Here we study the relationship between the diameter D of A and the least period T of B. We show that T is at most C exp(C \sqrt D \log D \sqrt{\log\log D}) and that we can have T at least quadratic in D.
Recently there has been significant progress in classifying integer friezes and $\text{SL}_2$-tilings. Typically, combinatorial methods are employed, involving triangulations of regions and inventive counting techniques. Here we develop a unified approach to such classifications using the tessellation of the hyperbolic plane by ideal triangles induced by the Farey graph. We demonstrate that the geometric, numeric and combinatorial properties of the Farey graph are perfectly suited to classifying tame $\text{SL}_2$-tilings, positive integer $\text{SL}_2$-tilings, and tame integer friezes -- bot…
The notion of $SL_2$-tiling is a generalization of that of classical Coxeter-Conway frieze pattern. We classify doubly antiperiodic $SL_2$-tilings that contain a rectangular domain of positive integers. Every such $SL_2$-tiling corresponds to a pair of frieze patterns and a unimodular $2\times2$-matrix with positive integer coefficients. We relate this notion to triangulated $n$-gons in the Farey graph.
A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some $M\in\mathbb{N}$ and $B\subset\mathbb{Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with $A$ and $B$. In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids, and saturating spaces, developed through a combination…
We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d > 2), which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension d=2 (the original Steinhaus problem) the question remains open.
Let k_1,...,k_d be positive integers, and D be a subset of [k_1]x...x[k_d], whose complement can be decomposed into disjoint sets of the form {x_1}x...x{x_{s-1}}x[k_s]x{x_{s+1}}x...x{x_d}. We conjecture that the number of elements of D can be represented as a linear combination of the numbers k_1,..., k_d with non-negative integer coefficients. A connexion of this conjecture with the structure of periodical cube tilings is revealed.
The set B_{p,r}^q:=\{\floor{nq/p+r} \colon n\in Z \} with integers p, q, r) is a Beatty set with density p/q. We derive a formula for the Fourier transform \hat{B_{p,r}^q}(j):=\sum_{n=1}^p e^{-2 πi j \floor{nq/p+r} / q}. A. S. Fraenkel conjectured that there is essentially one way to partition the integers into m>2 Beatty sets with distinct densities. We conjecture a generalization of this, and use Fourier methods to prove several special cases of our generalized conjecture.
Suppose $f\in L^1(\mathbb{R}^d)$, $Λ\subset\mathbb{R}^d$ is a finite union of translated lattices such that $f+Λ$ tiles with a weight. We prove that there exists a lattice $L\subset{\mathbb{R}}^d$ such that $f+L$ also tiles, with a possibly different weight. As a corollary, together with a result of Kolountzakis, it implies that any convex polygon that multi-tiles the plane by translations admits a lattice multi-tiling, of a possibly different multiplicity. Our second result is a new characterization of convex polygons that multi-tile the plane by translations. It also provides a very efficien…
Several discrete geometry problems are closely related to the arithmetic theory of elliptic curves defined on the rational fields $\mathbb{Q}$. In this paper we consider the $θ$-congruent number for $θ=\fracπ{3}$ and $\frac{2π}{3}$ and tiling number n. For the case that $n\geqslant 2$ is square-free odd integer, we determine all $n$ such that the Selmer rank of elliptic curve $E_{n,\fracπ{3}}:\ y^2=x(x-n)(x+3n)$ or/and $E_{n,\frac{2π}{3}}:\ y^2=x(x+n)(x-3n)$ is zero. From this, we provide several series of non $θ$-congruent numbers for $θ=\fracπ{3}$ and $\frac{2π}{3}$, and non tiling numbers n…
We consider "cubes" in products of finite cyclic groups and we study their tiling and spectral properties. (A set in a finite group is called a tile if some of its translates form a partition of the group and is called spectral if it admits an orhogonal basis of characters for the functions supported on the set.) We show an analog of a theorem due to Iosevich and Pedersen, Lagarias, Reeds and Wang, and the third author of this paper, which identified the tiling complements of the unit cube in Euclidean space with the spectra of the same cube.
Motivated by a question of Erdős, this paper considers questions concerning the discrete dynamical system on the 3-adic integers given by multiplication by 2. Let the 3-adic Cantor set consist of all 3-adic integers whose expansions use only the digits 0 and 1. The exception set is the set of 3-adic integers whose forward orbits under this action intersects the 3-adic Cantor set infinitely many times. It has been shown that this set has Hausdorff dimension 0. Approaches to upper bounds on the Hausdorff dimensions of these sets leads to study of intersections of multiplicative translates of Can…
The purpose of this paper is to investigate the properties of spectral and tiling subsets of cyclic groups, with an eye towards the spectral set conjecture in one dimension, which states that a bounded measurable subset of $\mathbb{R}$ accepts an orthogonal basis of exponentials if and only if it tiles $\mathbb{R}$ by translations. This conjecture is strongly connected to its discrete counterpart, namely that in every finite cyclic group, a subset is spectral if and only if it is a tile. The tools presented herein are refinements of recent ones used in the setting of cyclic groups; the structu…
