arXiv subjects

Itay Londner

Publications and source records attributed to Itay Londner.

10 recordsLinked to original sources

arXiv2024

Functional tilings and the Coven-Meyerowitz tiling conditions

Coven and Meyerowitz formulated two conditions which have since been conjectured to characterize all finite sets that tile the integers by translation. By periodicity, this conjecture is reduced to sets which tile a finite cyclic group $\mathbb{Z}_M$. In this paper we consider a natural relaxation of this problem, where we replace sets with nonnegative functions $f,g$, such that $f(0)=g(0)=1$, $f\ast g=\mathbf{1}_{\mathbb{Z}_M}$ is a functional tiling, and $f, g$ satisfy certain further natural properties associated with tilings. We show that the Coven-Meyerowitz tiling conditions do not necessarily hold in such generality. Such examples of functional tilings carry the potential to lead to proper tiling counterexamples to the Coven-Meyerowitz conjecture in the future.

math.CO↗

arXiv2024

A lonely weak tile

The notion of weak tiling was a key ingredient in the proof of Fuglede's spectral set conjecture for convex bodies \cite{conv}, due to the fact that every spectral set tiles its complement weakly with a suitable Borel measure. In this paper we review the concept of weak tiling, and answer a question raised in \cite{weak} by giving an example of a set $T$ which tiles its complement weakly, but $T$ is neither spectral, nor a proper tile.

math.CO↗

arXiv2024

Splitting for integer tilings

We consider translational integer tilings by finite sets $A\subset\mathbb{Z}$. We introduce a new method based on \emph{splitting}, together with a new combinatorial interpretation of some of the main tools from our earlier work. We also use splitting to prove the Coven-Meyerowitz conjecture for a new class of tilings $A\oplus B=\mathbb{Z}_M$. This includes tilings of period $M=p_1^{n_1}p_2^{n_2}p_3^{n_3}$ with $p_1>p_2^{n_2-1}p_3^{n_3-1}$, and tilings of period $M=p_1^{n_1}p_2^2p_3^2p_4^2$ with $p_1>p_2p_3p_4$, where $p_1,p_2,p_3,p_4$ are distinct primes and $n_1,n_2,n_3\in\mathbb{N}$. This is the second one of the two papers replacing version 1 of arXiv:2207.11809 (the first one is available as arXiv:2207.11809 v2). The main results of this paper (Theorem 1.2, Corollaries 1.4 and 1.5) and the intermediate results in Section 4.2 are all new and did not appear previously in arXiv:2207.11809 v1 or anywhere else. The material in Sections 3, 4.1, and 5 (splitting and the splitting formulation of the slab reduction) did appear in arXiv:2207.11809 v1 and has been removed from arXiv:2207.11809 v2. The results in Section 7 were included in arXiv:2207.11809 v1 and have been removed from arXiv:2207.11809 v2; the proofs are shorter and (we hope) more readable.

math.CO↗

arXiv2022

The Coven-Meyerowitz tiling conditions for 3 odd prime factors

It is well known that if a finite set $A\subset\mathbb{Z}$ tiles the integers by translations, then the translation set must be periodic, so that the tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. We are interested in characterizing all finite sets $A\subset\mathbb{Z}$ that have this property. Coven and Meyerowitz (1998) proposed conditions (T1), (T2) that are sufficient for $A$ to tile, and necessary when the cardinality of $A$ has at most two distinct prime factors. They also proved that (T1) holds for all finite tiles, regardless of size. It is not known whether (T2) must hold for all tilings with no restrictions on the number of prime factors of $|A|$. We prove that the Coven-Meyerowitz tiling condition (T2) holds for all integer tilings of period $M=(p_ip_jp_k)^2$, where $p_i,p_j,p_k$ are distinct odd primes. The proof also provides a classification of all such tilings.

math.CO↗

arXiv2022

The Coven-Meyerowitz tiling conditions for 3 prime factors: the even case

We consider finite sets $A\subset\mathbb{Z}$ tiles the integers by translations. By periodicity, any such tiling is equivalent to a factorization $A\oplus B=\mathbb{Z}_M$ of a finite cyclic group. Building on por previous work, we prove that a tentative characterization of finite tiles proposed by Coven and Meyerowitz holds for all integer tilings of period $M=(p_ip_jp_k)^2$, where $p_i,p_j,p_k$ are distinct primes. This extends the main result of [15] (Invent. Math. 2023), where we assumed that $M$ is odd. We also improve parts of the argument from [15]. We have split the earlier (70-page) version into two papers. The current version (49 pages) is the first of the two. The main result is the same as in the previous version: we prove (T2) in the 3-prime even case. The second paper will be posted shortly as a new submission. It will have a new main result where we prove (T2) for a new class of tilings (proved very recently, not included in v1 of this paper). Splitting-related results from the earlier 70-page version of this paper have been moved there.

math.CO↗

arXiv2022

Combinatorial and harmonic-analytic methods for integer tilings

A finite set of integers $A$ tiles the integers by translations if $\mathbb{Z}$ can be covered by pairwise disjoint translated copies of $A$. Restricting attention to one tiling period, we have $A\oplus B=\mathbb{Z}_M$ for some $M\in\mathbb{N}$ and $B\subset\mathbb{Z}$. This can also be stated in terms of cyclotomic divisibility of the mask polynomials $A(X)$ and $B(X)$ associated with $A$ and $B$. In this article, we introduce a new approach to a systematic study of such tilings. Our main new tools are the box product, multiscale cuboids, and saturating spaces, developed through a combination of harmonic-analytic and combinatorial methods. We provide new criteria for tiling and cyclotomic divisibility in terms of these concepts. As an application, we can determine whether a set $A$ containing certain configuration can tile a cyclic group $\mathbb{Z}_M$, or recover a tiling set based on partial information about it. We also develop tiling reductions where a given tiling can be replaced by one or more tilings with a simpler structure. The tools introduced here are crucial in our proof in a follow-up paper that all tilings of period $(pqr)^2$, where $p,q,r$ are distinct odd primes, satisfy a tiling condition proposed by Coven and Meyerowitz.

math.CO↗

arXiv2019

Optimal arithmetic structure in exponential Riesz sequences

We consider exponential systems $E\left(Λ\right)=\left\{ e^{iλt}\right\} _{λ\inΛ}$ for $Λ\subset\mathbb{Z}$. It has been shown by Londner and Olevskii in [9] that there exists a subset of the circle, of positive Lebesgue measure, so that every set Λwhich contains, for arbitrarily large N, an arithmetic progressions of length N and step $\ell=O\left(N^α\right)$, $α<1$, cannot be a Riesz sequence in the $L^{2}$ space over that set. On the other hand, every set admits a Riesz sequence containing arbitrarily long arithmetic progressions of length N and step $\ell=O\left(N\right)$. In this paper we show that every set $\mathcal{S}\subset\mathbb{T}$ of positive measure belongs to a unique class, defined through the optimal growth rate of the step of arithmetic progressions with respect to the length that can be found in Riesz sequences in the space $L^{2}\left(\mathcal{S}\right)$. We also give a partial geometric description of each class.

math.CA↗

arXiv2019

On syndetic Riesz sequences

Applying the solution to the Kadison-Singer problem, we show that every subset $\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\left\{ e^{iλx}\right\} _{λ\in Λ}$ such that $Λ\subset\mathbb{Z}$ is a set with gaps between consecutive elements bounded by ${\displaystyle \frac{C}{\left|\mathcal{S}\right|}}$. In the case when $\mathcal{S}$ is an open set we demonstrate, using quasicrystals, how such $Λ$ can be deterministically constructed.

math.CA↗

arXiv2017

Riesz sequences and generalized arithmetic progressions

The purpose of this note is to verify that the results attained in [6] admit an extension to the multidimensional setting. Namely, for subsets of the two dimensional torus we find the sharp growth rate of the step(s) of a generalized arithmetic progression in terms of its size which may be found in an exponential systems satisfying the Riesz sequence property.

math.CA↗

arXiv2016

Riesz sequences and arithmetic progressions

Given a set $\mathcal{S}$ of positive measure on the circle and a set of integers $Λ$, one may consider the family of exponentials $E\left(Λ\right):=\left\{ e^{iλt}\right\}_{λ\inΛ}$ and ask whether it is a Riesz sequence in the space $L^{2}\left(\mathcal{S}\right)$. We focus on this question in connection with some arithmetic properties of the set of frequencies. Improving a result of Bownik and Speegle, we construct a set $\mathcal{S}$ such that $E\left(Λ\right)$ is never a Riesz sequence if $Λ$ contains arbitrary long arithmetic progressions of length $N$ and step $\ell=O\left(N^{1-\varepsilon}\right)$. On the other hand, we prove that every set $\mathcal{S}$ admits a Riesz sequence $E\left(Λ\right)$ such that $Λ$ does contain arbitrary long arithmetic progressions of length $N$ and step $\ell=O\left(N\right)$.

math.CA↗