Tiling the integer lattice with translated sublattices

Combinatorial and harmonic-analytic methods for integer tilings

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…

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…

利用Bombieri-Siegel方法的离散与连续多重铺砌的精确不等式

Sharp inequalities for discrete and continuous multi-tiling, using the Bombieri-Siegel approach

该研究通过推导离散与改进连续版本的Bombieri-Siegel公式，得到了整数格与欧氏空间中多重铺砌的新等价条件，并推导出有界可测集体积相关的新谱公式。