Thomas Fallon, Azita Mayeli, Dominick Villano
For p=5,7, we show that a subset \(E \subset \F_p^3\) is spectral if and only if E tiles \(\F_p^3\) by translation. Additionally, we give an alternate proof that the conjecture holds for p=3.
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Fuglede's conjecture states that a subset $Ω\subseteq\mathbb{R}^{n}$ of positive and finite Lebesgue measure is a spectral set if and only if it tiles $\mathbb{R}^{n}$ by translation. The conjecture does not hold in both directions for $\mathbb{R}^n$, $n\ge3$. However, this conjecture remains open in $\mathbb{R}$ and $\mathbb{R}^2$. Cyclic groups play important roles in the study of Fuglede's conjecture in $\mathbb{R}$. In this paper, we introduce a new tool to study the spectral sets in cyclic groups. In particular, we prove that Fuglede's conjecture holds in $\mathbb{Z}_{p^{n}qr}$.
Fuglede's conjecture states that for a subset $Ω$ of a locally compact abelian group $G$ with positive and finite Haar measure, there exists a subset of the dual group of $G$ which is an orthogonal basis of $L^{2}(Ω)$ if and only if it tiles the group by translation. In this paper, we prove a divisibility property for a set in $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$. Then using the divisibility property and equi-distributed property, we prove that Fuglede's conjecture holds in the group $\mathbb{Z}_{p}\times\mathbb{Z}_{p^{n}}$.
We give an example of a set $Ω\subset \R^5$ which is a finite union of unit cubes, such that $L^2(Ω)$ admits an orthonormal basis of exponentials $\{\frac{1}{|Ω|^{1/2}} e^{2πi ξ_j \cdot x}: ξ_j \in Λ\}$ for some discrete set $Λ\subset \R^5$, but which does not tile $\R^5$ by translations. This answers a conjecture of Fuglede in the negative, at least in 5 and higher dimensions.
In this note we give an example of a set $\W\subset \R^4$ such that $L^2(\W)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\W |^{1/2}}e^{2πi x, ξ}\}_{ξ\inŁ}$ for some set $Ł\subset\R^4$, but which does not tile $\R^4$ by translations. This improves Tao's recent 5-dimensional example, and shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved.
We describe computer programs accompanying our paper and show that running them suffices to verify Fuglede's conjecture in $\mathbb{Z}_2^5$ and $\mathbb{Z}_2^6$.
Both directions of Fuglede's conjecture fail in dimension two
研究富格莱德猜想在二维空间的情况，通过在二阶有限阿贝尔群中构造特定子集，利用有限到无限转移原理得到\(\R^2\)中有界子集，证明该猜想在二维空间中两个方向都不成立。
The Fuglede conjecture states that a set is spectral if and only if it tiles by translation. The conjecture was disproved by T. Tao for dimensions 5 and higher by giving a counterexample in $\mathbb{Z}_3^5$. We present a computer program that determines that the Fuglede conjecture holds in $\mathbb{Z}_5^3$ by exhausting the search space. A. Iosevich, A. Mayeli and J. Pakianathan showed that the Fuglede conjecture holds over prime fields when the dimension does not exceed 2. The question for dimension 3 was previously addressed by Aten et al. for $p=3$. In this paper we build upon the results o…
In this short note we construct a spectral, non-tiling set of size $2p$ in $(\mathbb{Z}/p\mathbb{Z})^4$, $p$ odd prime. This example complements a previous counterexample in [arXiv:1509.01090], which existed only for $p \equiv 3 \pmod{4}$. On the contrary we show that the conjecture does hold in $(\mathbb{Z}/2\mathbb{Z})^4$.
Fuglede's conjecture on orthogonal bases of exponentials
本文回顾Fuglede 1974年论文中提出的猜想：欧几里得空间中哪些集合的L^2空间存在指数正交基，并讨论其相关内容。
We prove that a union of two intervals in R is a spectral set if and only if it tiles R by translations.
We present an approach to Fuglede's conjecture in $\mathbb{Z}_p^3$ using linear programming bounds, obtaining the following partial result: if $A\subseteq\mathbb{Z}_p^3$ with $p^2-p\sqrt{p}+\sqrt{p}<|A|<p^2$, then $A$ is not spectral.
Fuglede's conjecture holds for three intervals
研究实直线上既是谱集又是三个区间之并的有界可测子集，通过基于久期行列式的新余因子刚性方法，证明其能平移平铺直线，完成富格莱德猜想此类情况。
Let $Ω$ be a compact convex domain in the plane. We prove that $L^2(Ω)$ has an orthogonal basis of exponentials if and only if $Ω$ tiles the plane by translation.
Fuglede's spectral set conjecture states that a subset $Ω$ of a locally compact abelian group $G$ tiles the group by translation if and only if there exists a subset of continuous group characters which is an orthogonal basis of $L^2(Ω)$. We prove that Fuglede's conjecture holds on cyclic groups $\mathbb{Z}_{pqr}$ with $p,q,r$ distinct primes.
The study of Fuglede's conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjecture holds. For $\mathbb{Z}_p^3$ the problem is still open if $p\ge 11$. In connection we prove that Fuglede's conjecture holds on $\mathbb{Z}_{p}^2\times \mathbb{Z}_q$ by developing a method based on ideas from discrete geometry.
In this paper we study subsets $E$ of ${\Bbb Z}_p^d$ such that any function $f: E \to {\Bbb C}$ can be written as a linear combination of characters orthogonal with respect to $E$. We shall refer to such sets as spectral. In this context, we prove the Fuglede Conjecture in ${\Bbb Z}_p^2$ which says that $E \subset {\Bbb Z}_p^2$ is spectral if and only if $E$ tiles ${\Bbb Z}_p^2$ by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof.
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.
Fuglede's conjecture in $\mathbb{Q}_p$ is proved. That is to say, a Borel set of positive and finite Haar measure in $\mathbb{Q}_p$ is a spectral set if and only if it tiles $\mathbb{Q}_p$ by translation.
The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups of square-free order, initiated by Laba and Meyerowitz, but the spectral-tile direction is far from being well-understood. The product of at most three primes as the order of the cyclic group was studied intensely in the last couple of years. In this paper we study the case when the order of the cyclic group is the product of four different primes and prove that Fuglede's conjecture holds in this case.
We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the discrete Pompeiu property. Using this description and the revealed connection we prove that Fuglede's conjecture holds for $\mathbb{Z}_{p^n q^2}$, where $p$ and $q$ are different primes. In particular, we show that every spectral subset of $\mathbb{Z}_{p^n q^2}$ tiles the group. Further, using our combinatorial methods we give a simple proof for the statement…
Recent methods developed by Tao \cite{tao}, Kolountzakis and Matolcsi \cite{nspec} have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in $\RR^5$ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a non-spectral tile. In search of lower dimensional non-spectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang \cite{lagwang}. In particular, we prove here that the USC and the ``tile $\to$ spectral'' direction of Fuglede's conjecture are equivalent in any dimens…
We show that the spectral set conjecture by Fuglede holds in the setting of cyclic groups of order $p^n q$, where $p$, $q$ are distinct primes and $n\geq1$. This means that a subset $E$ of such a group $G$ tiles the group by translation ($G$ can be partitioned into translates of $E$) if and only if there exists an orthogonal basis of $L^2(E)$ consisting of group characters. The main ingredient of the present proof is the structure of vanishing sums of roots of unity of order $N$, where $N$ has at most two prime divisors; the extension of this proof to the case of cyclic groups of order $p^n q^…
Fuglede's conjecture in $\mathbb{Z}_{p}^{d}$, $p$ a prime, says that a subset $E$ tiles $\mathbb{Z}_{p}^{d}$ by translation if and only if $E$ is spectral, meaning any complex-valued function $f$ on $E$ can be written as a linear combination of characters orthogonal with respect to $E$. We disprove Fuglede's conjecture in $\mathbb{Z}_{p}^{4}$ for all odd primes $p$, by using log-Hadamard matrices to exhibit spectral sets of size $2p$ which do not tile, extending the result of Aten et al. that the conjecture fails in $\mathbb{Z}_{p}^{4}$ for primes $p \equiv 3 \pmod 4$ and in $\mathbb{Z}_{p}^{5…
Let $Ω$ be a convex polytope in $\mathbb{R}^d$. We say that $Ω$ is spectral if the space $L^2(Ω)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that $Ω$ is spectral if and only if it can tile the space by translations. It is known that if $Ω$ tiles then it is spectral, but the converse was proved only in dimension $d=2$, by Iosevich, Katz and Tao. By a result due to Kolountzakis, if a convex polytope $Ω\subset \mathbb{R}^d$ is spectral, then it must be centrally symmetric. We prove that also all the facets of $Ω$ are c…
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. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $Ω= A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result i…
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 study Fuglede's conjecture on the direct product of two cyclic groups. After collecting the known results we prove that Fuglede's conjecture holds on $\mathbb{Z}_{pq} \times \mathbb{Z}_{pq} \cong \mathbb{Z}_p^2 \times \mathbb{Z}_q^2$.
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。
A conjecture of Fuglede states that a bounded measurable set $Ω$ in space, of measure 1, can tile space by translations if and only if the Hilbert space $L^2(Ω)$ has an orthonormal basis consisting of exponentials. If $Ω$ has the latter property it is called spectral. We generalize a result of Fuglede, that a triangle in the plane is not spectral, proving that every non-symmetric convex domain is not spectral.
We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their $L^2$ space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups $\ZZ^d$ and $\RR^d$ (for $d \ge 5$) thus disproving one direction of the Spectral Set Conjecture of Fuglede. The other direction was recently disproved by Tao.
In this paper, we show an equi-disctributed property in $2$-dimensional finite abelian groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ where $p$ is a prime number. By using this equi-disctributed property, we prove that Fuglede's spectral set conjecture holds on groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$, namely, a set in $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ is a spectral set if and only if it is a tile.
A conjecture of Fuglede states that a bounded measurable set D, of measure 1, can tile space by translations if and only if the Hilbert space L^2(D) has an orthonormal basis consisting of exponentials exp(i 2 pi lambda x). If D has the latter property it is called spectral. Let D be a polytope with the following property: there is a direction u such that, of all the polytope faces perpendicular to u, the total area of the faces pointing in the positive u direction is more than the total area of the faces pointing in the negative u direction. It is almost obvious that such a polytope D cannot t…
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。
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…
Spectral sets in $\mathbb{Z}_{p^2qr}$ tile
本文证明了当 p、q、r 为素数时，Z_{p^2qr} 中每个谱集都可铺砌，并结合 Malikiosis 的最新结果推出 Fuglede 猜想在该群上成立。
A bounded measurable set $Ω\subset{\mathbb R}^d$ is called a spectral set if it admits some exponential orthonormal basis $\{e^{2πi \langleλ,x\rangle}: λ\inΛ\}$ for $L^2(Ω)$. In this paper, we show that in dimension one $d=1$, any spectrum $Λ$ with $0\inΛ$ of a spectral set $Ω$ with Lebesgue measure normalized to 1 must be rational. Combining previous results that spectrum must be periodic, the Fuglede's conjecture on ${\mathbb R}^1$ is now equivalent to the corresponding conjecture on all cyclic groups ${\mathbb Z}_{n}.$
Orthogonal Fourier Analysis on Domains
本文回顾了Fuglede（谱集）猜想近30年的发展，未涵盖有限群类结果、谱测度版本等方向，材料选取受个人因素影响，侧重描述性介绍并提出若干开放问题。
Brennan's conjecture holds for semigroups of holomorphic functions
本文针对全纯函数连续半群这一特殊情形，结合经典复分析技术与Békollé-Bonami权和积分算子谱的近期结果，给出了Brennan猜想的一个简短证明。
Very recently, E. H. Lieb and J. P. Solovej stated a conjecture about the constant of embedding between two Bergman spaces of the upper-half plane. A question in relation with a Werhl-type entropy inequality for the affine $AX+B$ group. More precisely, that for any holomorphic function $F$ on the upper-half plane $Π^+$, $$\int_{Π^+}|F(x+iy)|^{2s}y^{2s-2}dxdy\le \frac{π^{1-s}}{(2s-1)2^{2s-2}}\left(\int_{Π^+}|F(x+iy)|^2 dxdy\right)^s $$ for $s\ge 1$, and the constant $\frac{π^{1-s}}{(2s-1)2^{2s-2}}$ is sharp. We prove differently that the above holds whenever $s$ is an integer and we prove that …
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