Andrew Granville, Izabella Laba, Yang Wang
We consider the problem of characterizing finite sets which tile the integers by translations. Coven and Meyerowitz (J. Algebra 1999) found necessary and sufficient conditions for a finite set A to tile the integers under the assumption that |A| has at most 2 distinct prime factors. The purpose of this article is to settle, for the first time, certain three-prime cases.
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Reduction of integer tiles via CRT and base-p digits
本文通过中国剩余定理坐标中的基 p 数字切片约化，将 Coven-Meyerowitz 的整数平铺刻画推广到任意基数，证明其充分性。
Aperiodic tile sets from Sturmian lattices
研究基于二次斜率的斯图姆单词构造非周期平铺集，核心方法是给出显式算法，关键要素有斯图姆晶格和德洛内集的有界位移等价性，能生成基础缩放常数为实二次域单位的非周期平铺集。
Any bounded tile of the field $\mathbb{Q}_p$ of $p$-adic numbers is a compact open set up to a zero Haar measure set. In this note, we give a simple and direct proof of this fact.
Sturmian lattices and Aperiodic tile sets
本文提出基于二次斜率斯特鲁米安词构造非周期性瓷砖集的算法，并利用斯特鲁米安晶格和Delone集的有界位移等价性进行分类和构造。
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.
Characterization of self-affine tile digit sets on $\mathbb{R}^n$
本文通过引入斜积形式数字集，完全刻画了一维自相似瓦片的数字集结构，并证明当底数至多含两个素因子时，相应的自相似瓦片均为谱集。
Based on tiles and on the Coven-Meyerowitz property, we present some examples and some general constructions of spectral subsets of integers.
Integer tile and Spectrality of Cantor-Moran measures with equidifferent digit sets
本文研究带等差数字集的 Cantor-Moran 测度，刻画有限层数字和集为整数砖的充要条件，并分析其与谱测度之间的必要、非充分及在附加条件下的等价关系。
Metallic mean Wang tiles II: the dynamics of an aperiodic computer chip
本文研究一族新的非周期王氏瓷砖集，其动力学与金属均值相关，并构造了到二维环面的因子映射，推广了黄金比例情形。
Unsupported Cyclotomic Divisors in Three-Prime Integer Tilings
本文证明三素数整数平铺中存在无支撑分圆因子，刻画了平方周期情形，并给出最小模数及Apéry集构造。
Consider the sets of integers $A$ that avoid any arrangement of $g$ congruent $h$-subsets. Our findings refine and improve upon some results by Erdős and Harzheim about these sets.
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…
Integer Sets of Large Harmonic Sum Which Avoid Long Arithmetic Progressions
本研究给出数字受限整数集不含k项算术级数的条件，通过大规模搜索找到不含4项与10项算术级数且调和和优于传统贪心构造的整数集，可高效计算相关集合与调和和。
We study tiling and spectral sets in vector spaces over prime fields. The classical Fuglede conjecture in locally compact abelian groups says that a set is spectral if and only if it tiles by translation. This conjecture was disproved by T. Tao in Euclidean spaces of dimensions 5 and higher, using constructions over prime fields (in vector spaces over finite fields of prime order) and lifting them to the Euclidean setting. Over prime fields, when the dimension of the vector space is less than or equal to $2$ it has recently been proven that the Fuglede conjecture holds (see \cite{IMP15}). In t…
Overlapping substitutions and tilings
本文提出重叠替换的概念，通过非整数矩阵定义拼块频率，并证明其动力系统唯一测度性，同时构造了多种一维和高维例子。
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, …
Spectral Tile Direction in the Group $\mathbb{Z}_{p^2} \times \mathbb{Z}_{q^2} \times \mathbb{Z}_r$
针对满足$p^2q^2<r$的不同素数$p,q,r$，证明循环群$\boldsymbol{\times \times \times}$中所有谱集均为 tile，从而确定该群在对应条件下为 Fuglede 群。
We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.
The study of the additive volume of sets can be reduced to the case of one-dimensional sets. The exact values of the volume of extremal sets are given as a conjecture.
${\mathbf Q}$-independence and the construction of $B_h$-sets of integers and lattice points
本文通过Q向量空间构造方法，研究了整数和格点的B_h集的生成问题。
We consider sets of positive integers containing no sum of two elements in the set and also no product of two elements. We show that the upper density of such a set is strictly smaller than 1/2 and that this is best possible. Further, we also find the maximal order for the density of such sets that are also periodic modulo some positive integer.
A relationship between the Riemann zeta function and a density on integer sets is explored. Several properties of the examined density are derived.
We prove that Hilbert's Tenth Problem for a ring of integers in a number field K has a negative answer if K satisfies two arithmetical conditions (existence of a so-called division-ample set of integers and of an elliptic curve of rank one over K). We relate division-ample sets to arithmetic of abelian varieties.
Several measures for the density of sets of integers have been proposed, such as the asymptotic density, the Schnirelmann density, and the Dirichlet density. There has been some work in the literature on extending some of these concepts of density to higher dimensional sets of integers. In this work, we propose an extension of the Dirichlet density for sets of Gaussian integers and investigate some of its properties.
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