Mihail N. Kolountzakis, Izabella Laba
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.
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A set $Ω$ in a locally compact abelian group is called spectral if $L^2(Ω)$ has an orthogonal basis of group characters. An important problem, connected with the so-called Spectral Set Conjecture (saying that $Ω$ is spectral if and only if a collection of translates of $Ω$ can partition the group), is the question of whether the spectrality of a product set $Ω= A \times B$, in a product group, implies the spectrality of the factors $A$ and $B$. Recently Greenfeld and Lev proved that if $I$ is an interval and $Ω\subseteq {\mathbb R}^d$ then the spectrality of $I \times Ω$ implies the spectralit…
Let $\Q=[0,1)^d$ denote the unit cube in $d$-dimensional Euclidean space \Rd and let \T be a discrete subset of \Rd. We show that the exponentials $e_t(x):=exp(i2πtx)$, $t\in\T$ form an othonormal basis for $L^2(\Q)$ if and only if the translates $\Q+t$, $t\in\T$ form a tiling of \Rd.
Spectral sets and weak tiling
本文研究了谱集几何必要条件“弱平铺”的性质，并证明其在凸体、非凸多胞体、乘积域及正测度Cantor集上的应用，深化了对Fuglede猜想的理解。
In \cite{BCKM} it was shown that "Tiling implies Spectral" holds for a union of three intervals and the reverse implication was studied under certain restrictive hypotheses on the associated spectrum. In this paper, we reinvestigate the "Spectral implies Tiling" part of Fuglede's conjecture for the three interval case. We first show that the "Spectral implies Tiling" for two intervals follows from the simple fact that two distinct circles have at most two points of intersections. We then attempt this for the case of three intervals and except for one situation are able to prove "Spectral impli…
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.
We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.
On Tiling and Spectral Sets in $\mathbb Z_{p^2}\times\mathbb Z_{p^2}$
研究$\mathbb Z_{p^2}\times\mathbb Z_{p^2}$中铺砌与谱集的等价性，通过分析辛谱对的零集，揭示辛傅里叶变换与常规傅里叶变换零集的差异，扩展了基变换的自由度。
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…
Tiling, spectrality and aperiodicity of connected sets
本文研究了连通集的周期平铺猜想和Fuglede猜想，证明了在足够高的维度下这两个猜想对于连通集均不成立。
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。
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。
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…
Spectrality of Prime Size Tiles
本文证明了素数尺寸瓷砖在$\mathbb Z^d$中必为谱性，并展示了相关点集的铺砌与谱性性质。
A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $Ω$ is a spectral set if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it has long been known that for a convex body $Ω\subset \mathbb{R}^d$ the "tiling implies spectral" part of the conjecture is in fact true. To the contrary, the "spectral implies tiling" direction of the conjecture for convex bodies was proved only in $\mathbb{R}^2$, and also in $\mathbb{R}^3$ und…
In this paper we prove the "Tiling implies Spectral" part of Fuglede's paper for the case of three intervals. Then we prove the "Spectral implies Tiling" part of the conjecture for the case of three equal intervals as also when the intervals have lengths 1/2, 1/4, 1/4. For the general case we change our approach to get information on the structure of the spectrum for the n-interval case. Finally, we use symbolic computations on Mathematica, and prove this part of the conjecture with an additional assumption on the spectrum.
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.
We consider the cubic non-linear Schrödinger equation on general closed (compact without boundary) Riemannian surfaces. The problem is known to be locally well-posed in $H^s(M)$ for $s>1/2$. Global well-posedness for $s\geq 1$ follows easily from conservation of energy and standard arguments. In this work, we extend the range of global well-posedness to $s>2/3$. This generalizes, without any loss in regularity, a similar result on $\T^2$. The proof relies on the I-method of Colliander, Keel, Staffilani, Takaoka, and Tao, a semi-classical bilinear Strichartz estimate proved by the author, and s…
Let $K\subset \Bbb R^d$ be a set with positive and finite Lebesgue measure. Let $Λ=M(\Bbb Z^{2d})$ be a lattice in $\Bbb R^{2d}$ with density dens$(Λ)=1$. It is well-known that if $M$ is a diagonal block matrix with diagonal matrices $A$ and $B$, then $\mathcal G(|K|^{-1/2}χ_K, Λ)$ is an orthonormal basis for $L^2(\Bbb R^d)$ if and only if $K$ tiles both by $A(\Bbb Z^d)$ and $B^{-t}(\Bbb Z^d)$. However, there has not been any intensive study when $M$ is not a diagonal matrix. We investigate this problem for a large class of important cases of $M$. In particular, if $M$ is any lower block trian…
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