谱集与弱平铺

Tiling and spectral properties of near-cubic domains

We prove that is a measurable domain tiles R or R^2 by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1, and give an example showing that there is no analogue of the tiling result in dimensions 3 and higher.

弱平铺的几何意义

Geometric implications of weak tiling

本文重新审视弱平铺概念，证明了弱平铺补集的凸多胞体必须对称且具有对称面，并给出了实数轴上区间有限并集间隙长度的必要条件。

Spectrality and tiling by cylindric domains

A bounded set $Ω\subset \mathbb{R}^d$ is called a spectral set if the space $L^2(Ω)$ admits a complete orthogonal system of exponential functions. We prove that a cylindric set $Ω$ is spectral if and only if its base is a spectral set. A similar characterization is obtained of the cylindric sets which can tile the space by translations.

The structure of multiplicative tilings of the real line

Suppose $Ω, A \subseteq \RR\setminus\Set{0}$ are two sets, both of mixed sign, that $Ω$ is Lebesgue measurable and $A$ is a discrete set. We study the problem of when $A \cdot Ω$ is a (multiplicative) tiling of the real line, that is when almost every real number can be uniquely written as a product $a\cdot ω$, with $a \in A$, $ω\in Ω$. We study both the structure of the set of multiples $A$ and the structure of the tile $Ω$. We prove strong results in both cases. These results are somewhat analogous to the known results about the structure of translational tiling of the real line. There is, h…

Tiling sets and spectral sets over finite fields

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…

Simultaneous dilation and translation tilings of $\mathbb R^n$

We solve the wavelet set existence problem. That is, we characterize the full-rank lattices $Γ\subset \mathbb R^n$ and invertible $n \times n$ matrices $A$ for which there exists a measurable set $W$ such that $\{W + γ: γ\in Γ\}$ and $\{A^j(W): j\in \mathbb Z\}$ are tilings of $\mathbb R^n$. The characterization is a non-obvious generalization of the one found by Ionascu and Wang, which solved the problem in the case $n = 2$. As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showin…

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, …

用函数平铺p进数域$\n\mathbb{Q}_p$

Tiling the field $\mathbb{Q}_p$ of $p$-adic numbers by a function

本研究探讨p进数域中可平移平铺函数的性质，证明其一致局部常值性，解答Leptin与Müller的问题并给出离散集对应一致单位分解的充要条件，关联Fuglede猜想，刻画乘积空间平铺集结构并证明其为谱集。