The study of translational tiling with Fourier Analysis

The structure of translational tilings in $\mathbb{Z}^d$

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, …

平移铺砌：结构化还是狂野？

Translational tilings: structured or wild?

本文综述平移铺砌的结构研究及近期进展，聚焦集合经平移无间隙、无重叠填充空间时刚性结构与“狂野”行为的分界，并展现其跨学科特征。

Periodic structure of translational multi-tilings in the plane

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…

关于平移铺砌的傅里叶解析准则的一个例子

An example concerning Fourier analytic criteria for translational tiling

本文构造了一个离散集Λ（整数的小扰动）的反例，证明存在傅里叶变换零集相同但平移铺砌性质不同的函数，表明此类傅里叶解析准则无法完全刻画该集合的铺砌特性。

On the structure of multiple translational tilings by polygonal regions

We consider polygons with the following ``pairing property'': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon $K$ tiles the plane multiply when translated at the locations $Λ$, where $Λ$ is a multiset in the plane. The pairing property of $K$ makes this question particularly amenable to Fourier Analysis. After establishing a necessary and sufficient condition for $K$ to tile with a given lattice $Λ$ (which was first found by Bolle for the case of convex polygons-notice that all convex polygons that tile, necessarily hav…

$\boldsymbol{\text{R}}^n$中的平移自仿 tiles

Translational self-affine tiles in $\mathbb{R}^n$

该研究解决积分自仿 tile 理论中生成平移 tile 的数字集刻画问题，证明二维积分自仿 tile 存在完全周期 tiling，验证对应周期 tiling 猜想成立。