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Mihail N. Kolountzakis

Publications and source records attributed to Mihail N. Kolountzakis.

At least 19 recordsLinked to original sources
arXiv2026

Weak Tiling by Unions of Non-overlapping Unit Cubes

We study weak tiling by finite unions of pairwise non-overlapping unit cubes in $\mathbb R^d$ that are not necessarily axis-parallel. We obtain a weak-tiling analogue of Keller's classical theorem, which gives a structural restriction on the support of any weak tiling measure associated with the unit cube. This allows us to derive geometric restrictions on configurations of cubes whose union admits a weak tiling. As an application, we prove Fuglede's conjecture for unions of three non-overlapping unit squares in $\mathbb R^2$ as well as unions of two non-overlapping axis-parallel unit cubes in any dimension.

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arXiv2026

Bounded lattice tiles that pack with another lattice

Suppose $L, M$ are full-rank lattices in Euclidean space, such that ${\mathrm{vol}} L < {\mathrm{vol}} M$. Answering a question of Han and Wang from 2001, we show how to construct a bounded measurable set $F$ (we can even take $F$ to be a finite union of polytopes) such that $F+L$ is a tiling and $F+M$ is a packing. In the related problem of finding a common fundamental domain for two lattices $L$ and $M$ we show that no such common fundamental domain exists if the volumes are different even, if we do not require measurability of this domain.

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arXiv2026

Bounded common fundamental domains for two lattices

We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such that $E$ tiles $\mathbb{R}^d$ when translated by $L$ or by $M$. In fact, the set $E$ can be taken to be a finite union of polytopes. A consequence of this is that the indicator function of $E$ forms a Weyl--Heisenberg (Gabor) orthogonal basis of $L^2(\mathbb{R}^d)$ when translated by $L$ and modulated by $M^*$, the dual lattice of $M$.

math.CA↗
arXiv2026

Functions with comparable integrals on all k-planes

Let $d \ge 2$ and $1 \le k \le d-1$. We show that if $f:\mathbb{R}^d\to\mathbb{R}$ is nonnegative and measurable and $0<m\le M <\infty$ then it is impossible that on almost all affine $k$-planes $P$ in $\mathbb{R}^d$ the integral of $f$ on $P$ lies between $m$ and $M$. Let $G = \mathbb{Z}^k \times \{0\}^{d-k}$. Using $m=M=1$ and $f$ being the indicator function of a measurable set in $\mathbb{R}^d$ this then implies that there is no measurable Steinhaus set for the group $G$ in $\mathbb{R}^d$. In other words there is no measurable set $S \subseteq \mathbb{R}^d$ such that $S$ tiles $\mathbb{R}^d$ with $T(G)$, for all $T \in O(d)$. We also generalize our impossibility results concerning the size of line-integrals of functions to integrals on strips in the plane.

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arXiv2026

Tilings, packings, and the existence of Schwartz-class Gabor windows

The existence and construction of Schwartz-class windows for lattice Gabor frames is a central problem in time--frequency analysis. For general time--frequency lattices, we characterize exactly those lattices that admit Schwartz-class windows and, additionally, windows in the Feichtinger algebra. Thereby we arrive at sharp classical and amalgam Balian--Low theorems for lattices in $\mathbb{R}^d$. For separable lattices and specified block-zero forms, we construct smooth windows that are compactly supported either in time or in frequency. These constructions are based on a characterization of pairs of lattices for which there exists a single set that tiles with one lattice and whose $\varepsilon$-neighborhood packs with the other.

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arXiv2026

Spectrality of factors of product spectral measures

We refine the method by Greenfeld and Lev for the product spectral set problem and generalize the theorem to a singular measure setting. Furthermore, we establish a new class of spectral unions of intervals for which the product spectral set question has a positive answer. More precisely, if $A$ is a subset of the natural numbers such that $A\oplus B = \{0,1,\cdots, N-1\}$ for some $B\subset \mathbb N$ and $N>1$ then the product measure $\mathcal{L}|_{A+[0,1]}\times ν$ is a spectral measure (that may be singular) if and only if $ν$ is a spectral measure.

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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↗
arXiv2025

Spectra for finite unions of line segments

In this paper we study the spectrality of arc-length measures supported on the union of two line segments in the plane. We show that any such spectral measure must admit a line spectrum. Moreover, when the two segments are non-parallel, such spectral measure admits only line spectra. Thus, in this case every spectrum is one dimensional. In addition we show that this property fails for unions of three or more segments in the plane. We construct some arc-length spectral measures supported on the union of at least three line segments such that none of its spectra is contained in a line. Finally, we work in the general framework of arc-length measures supported on finite unions of curves in $\mathbb{R}^d$. We show that the size of any orthogonal set for such a measure inside a ball of radius $R$ grows at most linearly in $R$. We also give an alternative proof of this bound, and in fact obtain a more general result of growth rate of orthogonal sets for Ahlfors--David regular measures in $\mathbb{R}^d$ (not restricted to the one-dimensional setting).

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arXiv2025

Exponential polynomials and identification of polygonal regions from Fourier samples

Consider the set $E(D, N)$ of all bivariate exponential polynomials $$ f(ξ, η) = \sum_{j=1}^n p_j(ξ, η) e^{2πi (x_jξ+y_jη)}, $$ where the polynomials $p_j \in \mathbb{C}[ξ, η]$ have degree $<D$, $n\le N$ and where $x_j, y_j \in \mathbb{T} = \mathbb{R}/\mathbb{Z}$. We find a set $A \subseteq \mathbb{Z}^2$ that depends on $N$ and $D$ only and is of size $O(D^2 N \log N)$ such that the values of $f$ on $A$ determine $f$. Notice that the size of $A$ is only larger by a logarithmic quantity than the number of parameters needed to write down $f$. We use this in order to prove some uniqueness results about polygonal regions given a small set of samples of the Fourier Transform of their indicator function. If the number of different slopes of the edges of the polygonal region is $\le k$ then the region is determined from a predetermined set of Fourier samples that depends only on $k$ and the maximum number of vertices $N$ and is of size $O(k^2 N \log N)$. In the particular case where all edges are known to be parallel to the axes the polygonal region is determined from a set of $O(N \log N)$ Fourier samples that depends on $N$ only. Our methods are non-constructive.

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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.

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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.

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arXiv2025

Non-spectrality of some piecewise smooth curves and unions of line segments

We develop a systematic study about the spectrality of measures supported on piecewise smooth curves by studying the support of the tempered distributions arising from the tiling equation of some singular spectral measures. In doing so, we show that the arc-length measures of all closed polygonal lines are not spectral. {In particular, the boundary of a square is not spectral. We also show that the ``plus space'' (two crossing line segments) is not spectral.} Furthermore, our theory also shows that the arc length measures on {smooth} convex curves with finitely many transverse self-intersections are not spectral. Finally, several natural open questions about the spectrality of singular measures and {piecewise} smooth curves will also be discussed.

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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.

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arXiv2025

Spectrality of a measure consisting of two line segments

Take an interval $[t, t+1]$ on the $x$-axis together with the same interval on the $y$-axis and let $ρ$ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lai, Liu and Prince (2021) as well as Ai, Lu and Zhou (2023) we examine the spectrality of this measure for all different values of $t$ (being spectral means that there is an orthonormal basis for $L^2(ρ)$ consisting of exponentials $e^{2πi (λ_1 x + λ_2 y)}$). We almost complete the study showing that for $-\frac12<t<0$ and for all $t \notin {\mathbb Q}$ the measure $ρ$ is not spectral. The only remaining undecided case is the case $t=-\frac12$ (plus space). We also observe that in all known cases of spectral instances of this measure the spectrum is contained in a line and we give an easy necessary and sufficient condition for such measures to have a line spectrum.

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arXiv2024

Tiling, spectrality and aperiodicity of connected sets

Let $Ω\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $Ω$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's conjecture suggests that $Ω$ admits an orthogonal basis of exponential functions if and only if it tiles $\mathbb{R}^d$ by translations. Both conjectures are known to be false in sufficiently high dimensions, with all the so-far-known counterexamples being highly disconnected. On the other hand, both conjectures are known to be true for convex sets. In this work we study these conjectures for connected sets. We show that the periodic tiling conjecture, as well as both directions of Fuglede's conjecture are false for connected sets in sufficiently high dimensions.

math.CA↗
arXiv2024

Orthogonal Fourier Analysis on Domains

In this paper we go over the history of the Fuglede or Spectral Set Conjecture as it has developed over the last 30 years or so. We do not aim to be exhaustive and we do not cover important areas of development such as the results on the problem in classes of finite groups or the version of the problem that focuses on spectral measures instead of sets. The selection of the material has been strongly influenced by personal taste, history and capabilities. We are trying to be more descriptive than detailed and we point out several open questions.

math.CA↗
arXiv2024

Curves in the Fourier zeros of polytopal regions and the Pompeiu problem

We prove that any finite union $P$ of interior-disjoint polytopes in ${\mathbb R}^d$ has the Pompeiu property, a result first proved by Williams [Wil76]. This means that if a continuous function $f$ on $R^d$ integrates to 0 on any congruent copy of $P$ then $f$ is identically 0. By a fundamental result of Brown, Schreiber and Taylor [BST73] this is equivalent to showing that the Fourier-Laplace transform of the indicator function of $P$ does not vanish identically on any 0-centered complex sphere in ${\mathbb C}^d$ . Our proof initially follows the recent one of Machado and Robins [MR23] who are using the Brion-Barvinok formula for the Fourier-Laplace transform of a polytope. But we simplify this method considerably by removing the use of properties of Bessel function zeros. Instead we use some elementary arguments on the growth of linear combinations of exponentials with rational functions as coefficients. Our approach allows us to prove the non-existence of complex spheres of any center in the zero-set of the Fourier-Laplace transform. The planar case is even simpler in that we do not even need the Brion-Barvinok formula. We then go further in the question of which sets can be contained in the null set of the Fourier-Laplace transform of a polytope by extending results of Engel [Eng23] who showed that rationally parametrized hypersurfaces, under some mild conditions, cannot be contained in this null-set. We show that a rationally parametrized curve which is not contained in an affine hyperplane in ${\mathbb C}^d$ cannot be contained in this null-set. Results about curves parametrized by meromorphic functions are also given.

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