arXiv subjects

Nir Lev

Publications and source records attributed to Nir Lev.

At least 19 recordsLinked to original sources

arXiv2026

Bounded remainder sets, bounded distance equivalent cut-and-project sets, and equidecomposability

We use the measurable Hall's theorem due to Cieśla and Sabok to prove that (i) if two measurable sets $A,B \subset \mathbb{R}^d$ of the same measure are bounded remainder sets with respect to a totally irrational $d$-dimensional vector $α$, then $A, B$ are equidecomposable with measurable pieces using translations from $\mathbb{Z} α+ \mathbb{Z}^d$; and (ii) given a lattice $Γ\subset \mathbb{R}^m \times \mathbb{R}^n$ with projections $p_1$ and $p_2$ onto $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively, if two cut-and-project sets in $\mathbb{R}^m$ obtained from Riemann measurable windows $W, W' \subset \mathbb{R}^n$ are bounded distance equivalent, then $W, W'$ are equidecomposable with measurable pieces using translations from $p_2(Γ)$. We also prove by a different method that for one-dimensional cut-and-project sets, if the windows $W, W' \subset \mathbb{R}^n$ are polytopes then the pieces can also be chosen to be polytopes; however this result fails in dimensions two and higher.

math.MG↗

arXiv2026

Gabor unconditional bases and frames in $L^p(\mathbb{R})$

We consider the following problem: given a set $Λ\subset \mathbb{R} \times \mathbb{R}$ and $p \neq 2$, does there exist a function $g \in L^p(\mathbb{R})$ such that the Gabor system $\{g(x-t) e^{2 πisx}\}$, $(t,s) \in Λ$, consisting of time-frequency shifts of $g$, forms an unconditional basis or unconditional Schauder frame in the space $L^p(\mathbb{R})$? We completely resolve this question for $p>2$; in particular, we characterize the sets $Λ$ such that an unconditional Schauder frame of this form exists. We also prove a Balian-Low type result, showing that the window function $g$ cannot enjoy mild continuity and decay conditions. For $1<p<2$, we prove that a Gabor system cannot form an unconditional basis or unconditional Schauder frame in $L^p(\mathbb{R})$ if the set $Λ$ satisfies a natural separation condition.

math.CA↗

arXiv2026

Completeness of sparse, almost integer and finite local complexity sequences of translates in $L^p(\mathbb{R})$

A real sequence $Λ= \{λ_n\}_{n=1}^\infty$ is called $p$-generating if there exists a function $g$ whose translates $\{g(x-λ_n)\}_{n=1}^\infty$ span the space $L^p(\mathbb{R})$. While the $p$-generating sets were completely characterized for $p=1$ and $p>2$, the case $1 < p \le 2$ remains not well understood. In this case, both the size and the arithmetic structure of the set play an important role. In the present paper, (i) We show that a $p$-generating set $Λ$ of positive real numbers can be very sparse, namely, the ratios $λ_{n+1} / λ_n$ may tend to $1$ arbitrarily slowly; (ii) We prove that every "almost integer" sequence $Λ$, i.e. satisfying $λ_n = n + α_n$, $0 \neq α_n \to 0$, is $p$-generating; and (iii) We construct $p$-generating sets $Λ$ such that the successive differences $λ_{n+1} - λ_n$ attain only two different positive values. The constructions are, in a sense, sharp: it is well known that $Λ$ cannot be Hadamard lacunary and cannot be contained in any arithmetic progression.

math.CA↗

arXiv2025

Schauder frames of discrete translates in $L^2(\mathbb{R})$

We construct a uniformly discrete sequence $\{λ_1 < λ_2 < \cdots\} \subset \mathbb{R}$ and functions $g$ and $\{g_n^*\}$ in $L^2(\mathbb{R})$, such that every $f \in L^2(\mathbb{R})$ admits a series expansion \[ f(x) = \sum_{n=1}^{\infty} \langle f, g_n^* \rangle \, g(x-λ_n) \] convergent in the $L^2(\mathbb{R})$ norm.

math.CA↗

arXiv2025

Schauder frames of discrete translates in $L^p(\mathbb{R})$

For every $p > (1 + \sqrt{5})/2$ we construct a uniformly discrete real sequence $\{λ_n\}_{n=1}^\infty$ satisfying $|λ_n| = n + o(1)$, a function $g \in L^p(\mathbb{R})$, and continuous linear functionals $\{g^*_n\}_{n=1}^\infty$ on $L^p(\mathbb{R})$, such that every $f \in L^p(\mathbb{R})$ admits a series expansion \[ f(x) = \sum_{n=1}^{\infty} g_n^*(f) g(x-λ_n) \] convergent in the $L^p(\mathbb{R})$ norm. We moreover show that $g$ can be chosen nonnegative.

math.CA↗

arXiv2025

The Turán and Delsarte problems and their duals

We study two optimization problems for positive definite functions on Euclidean space with restrictions on their support and sign: the Turan problem and the Delsarte problem. These problems have been studied also for their connections to geometric problems of tiling and packing. In the finite group setting the weak and strong linear duality for these problems are automatic. We prove these properties in the continuous setting. We also show the existence of extremizers for these problems and their duals, and establish tiling-type relations between the extremal functions for each problem and the extremal measures or distributions for the dual problem. We then apply the results to convex bodies, and prove that the Delsarte packing bound is strictly better than the trivial volume packing bound for every convex body that does not tile the space.

math.CA↗

arXiv2025

Geometric implications of weak tiling

The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If $A \subset \mathbb{R}^d$ is a convex polytope, we give a direct and self-contained proof that $A$ must be symmetric and have symmetric facets. If $A \subset \mathbb{R}$ is a finite union of intervals, we give a necessary condition on the lengths of the gaps between the intervals.

math.CA↗

arXiv2025

Unconditional Schauder frames of exponentials and of uniformly bounded functions in $L^p$ spaces

It is known that there is no unconditional basis of exponentials in the space $L^p(Ω)$, $p \ne 2$, for any set $Ω\subset \mathbb{R}^d$ of finite measure. This is a consequence of a more general result due to Gaposhkin, who proved that the space $L^p(Ω)$ does not admit a seminormalized unconditional basis consisting of uniformly bounded functions. We show that the latter result fails if the word "basis" is replaced with "Schauder frame". On the other hand we prove that if $Ω$ has nonempty interior then there are no unconditional Schauder frames of exponentials in the space $L^p(Ω)$, $p \ne 2$.

math.CA↗

arXiv2025

Maximality and completeness of orthogonal exponentials on the cube

It is possible to have a packing by translates of a cube that is maximal (i.e.\ no other cube can be added without overlapping) but does not form a tiling. In the long running analogy of packing and tiling to orthogonality and completeness of exponentials on a domain, we pursue the question whether one can have maximal orthogonal sets of exponentials for a cube without them being complete. We prove that this is not possible in dimensions 1 and 2, but is possible in dimensions 3 and higher. We provide several examples of such maximal incomplete sets of exponentials, differing in size, and we raise relevant questions. We also show that even in dimension $1$ there are sets which are spectral (i.e. have a complete set of orthogonal exponentials) and yet they also possess maximal incomplete sets of orthogonal exponentials.

math.CA↗

arXiv2025

Support of extremal doubly stochastic arrays

An $n \times m$ array with nonnegative entries is called doubly stochastic if the sum of its entries at each row is $m$ and at each column is $n$. The set of all $n \times m$ doubly stochastic arrays is a convex polytope with finitely many extremal points. The main result of this paper characterizes the possible sizes of the supports of all extremal $n \times m$ doubly stochastic arrays. In particular we prove that the minimal size of the support of an $n \times m$ doubly stochastic array is $n + m - \gcd(n,m)$. Moreover, for $m=kn+1$ we also characterize the structure of the support of the extremal arrays.

math.CO↗

arXiv2025

Completeness of uniformly discrete translates in $L^p(\mathbb{R})$

We construct a real sequence $\{λ_n\}_{n=1}^{\infty}$ satisfying $λ_n = n + o(1)$, and a Schwartz function $f$ on $\mathbb{R}$, such that for any $N$ the system of translates $\{f(x - λ_n)\}$, $n > N$, is complete in the space $L^p(\mathbb{R})$ for every $p>1$. The same system is also complete in a wider class of Banach function spaces on $\mathbb{R}$.

math.CA↗

arXiv2024

There are no unconditional Schauder frames of translates in $L^p(\mathbb{R})$, $1 \le p \le 2$

It is known that a system formed by translates of a single function cannot be an unconditional Schauder basis in the space $L^p(\mathbb{R})$ for any $1 \le p < \infty$. To the contrary, there do exist unconditional Schauder frames of translates in $L^p(\mathbb{R})$ for every $p>2$. The existence of such a system for $1 < p \leq 2$, however, has remained an open problem. In this paper the problem is solved in the negative: we prove that none of the spaces $L^p(\mathbb{R})$, $1 \le p \le 2$, admits an unconditional Schauder frame of translates.

math.CA↗

arXiv2023

Spectral sets and weak tiling

A set $Ω\subset \mathbb{R}^d$ is said to be spectral if the space $L^2(Ω)$ admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that $Ω$ is spectral if and only if it can tile the space by translations. While this conjecture was disproved for general sets, it was recently proved that the Fuglede conjecture does hold for the class of convex bodies in $\mathbb{R}^d$. The proof was based on a new geometric necessary condition for spectrality, called "weak tiling". In this paper we study further properties of the weak tiling notion, and present applications to convex bodies, non-convex polytopes, product domains and Cantor sets of positive measure.

math.CA↗

arXiv2023

Functions tiling simultaneously with two arithmetic progressions

We consider measurable functions $f$ on $\mathbb{R}$ that tile simultaneously by two arithmetic progressions $α\mathbb{Z}$ and $β\mathbb{Z}$ at respective tiling levels $p$ and $q$. We are interested in two main questions: what are the possible values of the tiling levels $p,q$, and what is the least possible measure of the support of $f$? We obtain sharp results which show that the answers depend on arithmetic properties of $α, β$ and $p,q$, and in particular, on whether the numbers $α, β$ are rationally independent or not.

math.CA↗

arXiv2022

The Fuglede conjecture for convex domains is true in all dimensions

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 $\mathbb{R}^3$ under the a priori assumption that $Ω$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $Ω$ is a polytope) and could not be treated using the previously developed techniques. In this paper we fully settle Fuglede's conjecture for convex bodies affirmatively in all dimensions, i.e. we prove that if a convex body $Ω\subset \mathbb{R}^d$ is a spectral set then $Ω$ is a convex polytope which can tile the space by translations. To prove this we introduce a new technique, involving a construction from crystallographic diffraction theory, which allows us to establish a geometric "weak tiling" condition necessary for a set $Ω\subset \mathbb{R}^d$ to be spectral.

math.CA↗

arXiv2022

Gabor orthonormal bases, tiling and periodicity

We show that if the Gabor system $\{ g(x-t) e^{2πi s x}\}$, $t \in T$, $s \in S$, is an orthonormal basis in $L^2(\mathbb{R})$ and if the window function $g$ is compactly supported, then both the time shift set $T$ and the frequency shift set $S$ must be periodic. To prove this we establish a necessary functional tiling type condition for Gabor orthonormal bases which may be of independent interest.

math.CA↗

arXiv2021

Poisson summation formulas involving the sum-of-squares function

We obtain new Poisson type summation formulas with nodes $\pm \sqrt{n}$ and with weights involving the function $r_k(n)$ that gives the number of representations of a positive integer $n$ as the sum of $k$ squares. Our results extend summation formulas due to Guinand and Meyer that involve the sum-of-three-squares function $r_3(n)$.

math.CA↗

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↗