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Nir Lev

Publications and source records attributed to Nir Lev.

At least 37 records · Page 2Linked to original sources
arXiv2021

An example concerning Fourier analytic criteria for translational tiling

It is well-known that the functions $f \in L^1(\mathbb{R}^d)$ whose translates along a lattice $Λ$ form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a discrete set $Λ\subset \mathbb{R}$ (a small perturbation of the integers) for which no characterization of this kind is possible: there are two functions $f, g \in L^1(\mathbb{R})$ whose Fourier transforms have the same set of zeros, but such that $f + Λ$ is a tiling while $g + Λ$ is not.

math.CA↗
arXiv2021

Tiling by translates of a function: results and open problems

We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $Λ\subset \mathbb{R}$, if we have $\sum_{λ\in Λ} f(x-λ)=w$ a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of $\mathbb{R}$ by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.

math.CA↗
arXiv2020

Riesz bases of exponentials for convex polytopes with symmetric faces

We prove that for any convex polytope $Ω\subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(Ω)$. The result is new in all dimensions $d$ greater than one.

math.CA↗
arXiv2019

Spectrality of polytopes and equidecomposability by translations

Let $A$ be a polytope in $\mathbb{R}^d$ (not necessarily convex or connected). We say that $A$ is spectral if the space $L^2(A)$ has an orthogonal basis consisting of exponential functions. A result due to Kolountzakis and Papadimitrakis (2002) asserts that if $A$ is a spectral polytope, then the total area of the $(d-1)$-dimensional faces of $A$ on which the outward normal is pointing at a given direction, must coincide with the total area of those $(d-1)$-dimensional faces on which the outward normal is pointing at the opposite direction. In this paper, we prove an extension of this result to faces of all dimensions between $1$ and $d-1$. As a consequence we obtain that any spectral polytope $A$ can be dissected into a finite number of smaller polytopes, which can be rearranged using translations to form a cube.

math.CA↗
arXiv2019

Multi-tiling and equidecomposability of polytopes by lattice translates

We characterize the polytopes in $\mathbb{R}^d$ (not necessarily convex or connected ones) which multi-tile the space by translations along a given lattice. We also give a necessary and sufficient condition for two polytopes in $\mathbb{R}^d$ to be equidecomposable by lattice translations.

math.CO↗
arXiv2018

Spectrality of product domains and Fuglede's conjecture for convex polytopes

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 in connection with the conjecture that a convex polytope $Ω$ is spectral if and only if it can tile by translations.

math.CA↗
arXiv2017

Fuglede's spectral set conjecture for convex polytopes

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 centrally symmetric. These conditions are necessary for $Ω$ to tile by translations. We also develop an approach which allows us to prove that in dimension $d=3$, any spectral convex polytope $Ω$ indeed tiles by translations. Thus we obtain that Fuglede's conjecture is true for convex polytopes in $\mathbb{R}^3$.

math.CA↗
arXiv2017

Fourier quasicrystals and discreteness of the diffraction spectrum

We prove that a positive-definite measure in $\mathbb{R}^n$ with uniformly discrete support and discrete closed spectrum, is representable as a finite linear combination of Dirac combs, translated and modulated. This extends our recent results where we proved this under the assumption that also the spectrum is uniformly discrete. As an application we obtain that Hof's quasicrystals with uniformly discrete diffraction spectra must have a periodic diffraction structure.

math.CA↗
arXiv2017

Fourier frames for singular measures and pure type phenomena

Let $μ$ be a positive measure on $R^d$. It is known that if the space $L^2(μ)$ has a frame of exponentials then the measure $μ$ must be of "pure type": it is either discrete, absolutely continuous or singular continuous. It has been conjectured that a similar phenomenon should be true within the class of singular continuous measures, in the sense that $μ$ cannot admit an exponential frame if it has components of different dimensions. We prove that this is not the case by showing that the sum of an arc length measure and a surface measure can have a frame of exponentials. On the other hand we prove that a measure of this form cannot have a frame of exponentials if the surface has a point of non-zero Gaussian curvature. This is in spite of the fact that each "pure" component of the measure separately may admit such a frame.

math.CA↗
arXiv2016

Riesz bases, Meyer's quasicrystals, and bounded remainder sets

We consider systems of exponentials with frequencies belonging to simple quasicrystals in $\mathbb{R}^d$. We ask if there exist domains $S$ in $\mathbb{R}^d$ which admit such a system as a Riesz basis for the space $L^2(S)$. We prove that the answer depends on an arithmetical condition on the quasicrystal. The proof is based on the connection of the problem to the discrepancy of multi-dimensional irrational rotations, and specifically, to the theory of bounded remainder sets. In particular it is shown that any bounded remainder set admits a Riesz basis of exponentials. This extends to several dimensions (and to the non-periodic setting) the results obtained earlier in dimension one.

math.CA↗
arXiv2016

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.

math.CA↗
arXiv2015

On non-periodic tilings of the real line by a function

It is known that a positive, compactly supported function $f \in L^1(\mathbb R)$ can tile by translations only if the translation set is a finite union of periodic sets. We prove that this is not the case if $f$ is allowed to have unbounded support. On the other hand we also show that if the translation set has finite local complexity, then it must be periodic, even if the support of $f$ is unbounded.

math.CA↗
arXiv2015

Quasicrystals with discrete support and spectrum

We proved recently that a measure on R, whose support and spectrum are both uniformly discrete sets, must have a periodic structure. Here we show that this is not the case if the support and the spectrum are just discrete closed sets.

math.CA↗
arXiv2014

Sets of bounded discrepancy for multi-dimensional irrational rotation

We study bounded remainder sets with respect to an irrational rotation of the $d$-dimensional torus. The subject goes back to Hecke, Ostrowski and Kesten who characterized the intervals with bounded remainder in dimension one. First we extend to several dimensions the Hecke-Ostrowski result by constructing a class of $d$-dimensional parallelepipeds of bounded remainder. Then we characterize the Riemann measurable bounded remainder sets in terms of "equidecomposability" to such a parallelepiped. By constructing invariants with respect to this equidecomposition, we derive explicit conditions for a polytope to be a bounded remainder set. In particular this yields a characterization of the convex bounded remainder polygons in two dimensions. The approach is used to obtain several other results as well.

math.DS↗
arXiv2014

Separating signal from noise

Suppose that a sequence of numbers $x_n$ (a `signal') is transmitted through a noisy channel. The receiver observes a noisy version of the signal with additive random fluctuations, $x_n + ξ_n$, where $ξ_n$ is a sequence of independent standard Gaussian random variables. Suppose further that the signal is known to come from some fixed space of possible signals. Is it possible to fully recover the transmitted signal from its noisy version? Is it possible to at least detect that a non-zero signal was transmitted? In this paper we consider the case in which signals are infinite sequences and the recovery or detection are required to hold with probability one. We provide conditions on the signal space for checking whether detection or recovery are possible. We also analyze in detail several examples including spaces of Fourier transforms of measures, spaces with fixed amplitudes and the space of almost periodic functions. Many of our examples exhibit critical phenomena, in which a sharp transition is made from a regime in which recovery is possible to a regime in which even detection is impossible.

math.PR↗
arXiv2013

Equidistribution estimates for Fekete points on complex manifolds

We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.

math.CV↗