Fuglede's conjecture for a union of two intervals

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。

"Spectral implies Tiling" for Three Intervals Revisited

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…

When the algebraic difference of two central Cantor sets is an interval?

Let $C(a ),C(b)\subset \lbrack 0,1]$ be the central Cantor sets generated by sequences $ a,b \in (0,1)^{\mathbb{N}}$. The first main result of the paper gives a necessary and a sufficient condition for sequences $a$ and $b$ which inform when $C(a )-C(b)$ is equal to $[-1,1]$ or is a finite union of closed intervals. One of the corollaries following from this results shows that the product of thicknesses of two central Cantor sets which algebraic difference is an interval may be arbitrarily small. We also show that there are sets $C(a)$ and $C(b)$ with the Hausdorff dimension equal to $0$ such …

Spectrum is periodic for n-Intervals

In this paper we study spectral sets which are unions of finitely many intervals in R. We show that any spectrum associated with such a spectral set is periodic, with the period an integral multiple of the measure of the set. As a consequence we get a structure theorem for such spectral sets and observe that the generic case is that of the equal interval case.

On the One-Dimentional Pompeiu Problem

We investigate the Pompeiu property for subsets of the real line, under no assumption of connectedness. In particular we focus our study on finite unions of bounded (disjoint) intervals, and we emphasize the different results corresponding to the cases where the function in question is supposed to have constant integral on all isometric images, or just on all the translation-images of the domain. While no set of the previous kind enjoys the Pompeiu property in the latter sense, we provide a necessary and sufficient condition in order a union of two intervals to have the Pompeiu property in the…

等长三个区间之并集上的Turán问题

Turán problem on the union of three intervals of equal length

本文针对对称三区间集上的Turán极值问题，建立离散逼近原理将其转化为等价的有限半定规划问题，计算了整数和有理参数λ对应的Turán常数，分析了其对λ的依赖性并推导了λ>5时的上界。

Asymptotics for the Fredholm Determinant of the Sine Kernel on a Union of Intervals

In the bulk scaling limit of the Gaussian Unitary Ensemble of Hermitian matrices the probability that an interval of length $s$ contains no eigenvalues is the Fredholm determinant of the sine kernel $\sin(x-y)\overπ(x-y)$ over this interval. A formal asymptotic expansion for the determinant as $s$ tends to infinity was obtained by Dyson. In this paper we replace a single interval of length $s$ by $sJ$ where $J$ is a union of $m$ intervals and present a proof of the asymptotics up to second order. The logarithmic derivative with respect to $s$ of the determinant equals a constant (expressible i…

Combining Riesz bases

We show that any finite union of intervals supports a Riesz basis of exponentials

$L^2$上两个区间中的指数里斯基基

Exponential Riesz bases in $L^2$ on two interval

本文研究了两个区间上指数系统成为里斯基基的充分条件，并揭示了此类系统可能存在的一个额外点效应。

Some results related to the Logvinenko-Sereda theorem

We prove several results related to the theorem of Logvinenko and Sereda on determining sets for functions with Fourier transforms supported in an interval. We obtain a polynomial instead of exponential bound in this theorem, and we extend it to the case of functions with Fourier transforms supported in the union of a bounded number of intervals. The same results hold in all dimensions.

Subsum Sets: Intervals, Cantor Sets, and Cantorvals

Given a sequence converging to zero, we consider the set of numbers which are sums of (infinite, finite, or empty) subsequences. When the original sequence is not absolutely summable, the subsum set is an unbounded closed interval which includes zero. When it is absolutely summable the subsum set is one of the following: a finite union of (nontrivial) compact intervals, a Cantor set, or a "symmetric Cantorval".

Level Sets of Differentiable Functions of Two Variables with Non-vanishing Gradient

We show that if the gradient of $f:\RR^2\rightarrow\RR$ exists everywhere and is nowhere zero, then in a neighbourhood of each of its points the level set $\{x\in\RR^2:f(x)=c\}$ is homeomorphic either to an open interval or to the union of finitely many open segments passing through a point. The second case holds only at the points of a discrete set. We also investigate the global structure of the level sets.

Spectral properties of some unions of linear spaces

We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, with a natural additive measure $ρ$. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of $L^2(ρ)$ and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on ${\mathbb R}^1$. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We…

Riesz bases of exponentials on multiband spectra

Let $S$ be the union of finitely many disjoint intervals on the real line. Suppose that there are two real numbers $α, β$ such that the length of each interval belongs to $Z α+ Z β$. We use quasicrystals to construct a discrete set of real frequencies such that the corresponding system of exponentials is a Riesz basis in the space $L^2(S)$.

Geometry of limits of zeros of polynomial sequences of type $(1,2)$

We study the root distribution of some univariate polynomials satisfying a recurrence of order two with linear and quadratic polynomial coefficients. We show that the set of non-isolated limits of zeros of the polynomials is the closure of an arc, a circle, an interval or its exterior under the real line, the union of at most two different shapes of the above cases expect the union of an arc and a circle, and some degenerate forms.

Momentum Operators in Two Intervals: Spectra and Phase Transition

We study the momentum operator defined on the disjoint union of two intervals. Even in one dimension, the question of two non-empty open and non-overlapping intervals has not been worked out in a way that extends the cases of a single interval and gives a list of the selfadjoint extensions. Starting with zero boundary conditions at the four endpoints, we characterize the selfadjoint extensions and undertake a systematic and complete study of the spectral theory of the selfadjoint extensions. In an application of our extension theory to harmonic analysis, we offer a new family of spectral pairs…

On Hadamard powers of positive semi-definite matrices

Consider the set of scalars $α$ for which the $α$th Hadamard power of any $n\times n$ positive semi-definite (p.s.d.) matrix with non-negative entries is p.s.d. It is known that this set is of the form $\{0, 1, \dots, n-3\}\cup [n-2, \infty)$. A natural question is "what is the possible form of the set of such $α$ for a fixed p.s.d. matrix with non-negative entries?". In all examples appearing in the literature, the set turns out to be union of a finite set and a semi-infinite interval. In this article, examples of matrices are given for which the set consists of a finite set and more than one…

A vector equilibrium problem for symmetrically located point charges on a sphere

We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by r+1 equal point charges that are symmetrically located around the north pole. The support of the equilibrium measure is known as the droplet. The droplet has a motherbody which we characterize by means of a vector equilibrium problem (VEP) for r measures in the complex plane. The model undergoes two transitions which is reflected in the support of the first component of the minimizer of the VEP, namely the support can be a finite interval containing 0, the union of two intervals, or…