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Mihail N. Kolountzakis

Publications and source records attributed to Mihail N. Kolountzakis.

71 records · Page 4Linked to original sources
arXiv2004

Tiles with no spectra

We exhibit a subset of a finite Abelian group, which tiles the group by translation, and such that its tiling complements do not have a common spectrum (orthogonal basis for their $L^2$ space consisting of group characters). This disproves the Universal Spectrum Conjecture of Lagarias and Wang. Further, we construct a set in some finite Abelian group, which tiles the group but has no spectrum. We extend this last example to the groups $\ZZ^d$ and $\RR^d$ (for $d \ge 5$) thus disproving one direction of the Spectral Set Conjecture of Fuglede. The other direction was recently disproved by Tao.

math.CA↗
arXiv2003

Turán's extremal problem for positive definite functions on groups

We study the following question: Given an open set $Ω$, symmetric about 0, and a continuous, integrable, positive definite function $f$, supported in $Ω$ and with $f(0)=1$, how large can $\int f$ be? This problem has been studied so far mostly for convex domains $Ω$ in Euclidean space. In this paper we study the question in arbitrary locally compact abelian groups and for more general domains. Our emphasis is on finite groups as well as Euclidean spaces and $\ZZ^d$. We exhibit upper bounds for $\int f$ assuming geometric properties of $Ω$ of two types: (a) packing properties of $Ω$ and (b) spectral properties of $Ω$. Several examples and applications of the main theorems are shown. In particular we recover and extend several known results concerning convex domains in Euclidean space. Also, we investigate the question of estimating $\int_Ωf$ over possibly dispersed sets solely in dependence of the given measure $m:=|Ω|$ of $Ω$. In this respect we show that in $\RR$ and $\ZZ$ the integral is maximal for intervals.

math.CA↗
arXiv2003

A Weyl type formula for Fourier spectra and frames

We prove qualitative and quantitative results concerning the asymptotic density in dilates of centered convex bodies of the frequency vectors of orthogonal exponential bases and frames associated to bounded domains in Euclidean space.

math.CA↗
arXiv2003

Reconstruction of functions from their triple correlations

Suppose that A is a subset of an abelian group G. To know the 3-deck of A is to know the number of occurences in A of translates of each possible multiset {0,a,b}. The concept of the 3-deck is naturally extended to integrable functions on G. In this paper we study when the 3-deck of a function determines the function up to translations. The method is to look at the Fourier Transform of the function. Our emphasis is on the real line and the cyclic groups.

math.CA↗
arXiv2003

Distance sets corresponding to convex bodies

Suppose that $K \subseteq \RR^d$ is a 0-symmetric convex body which defines the usual norm $$ \Norm{x}_K = \sup\Set{t\ge 0: x \notin tK} $$ on $\RR^d$. Let also $A\subseteq\RR^d$ be a measurable set of positive upper density $ρ$. We show that if the body $K$ is not a polytope, or if it is a polytope with many faces (depending on $ρ$), then the distance set $$ D_K(A) = \Set{\Norm{x-y}_K: x,y\in A} $$ contains all points $t\ge t_0$ for some positive number $t_0$. This was proved by Katznelson and Weiss, by Falconer and Marstrand and by Bourgain in the case where $K$ is the Euclidean ball in any dimension. As corollaries we obtain (a) an extension to any dimension of a theorem of Iosevich and Łaba regarding distance sets with respect to convex bodies of well-distributed sets in the plane, and also (b) a new proof of a theorem of Iosevich, Katz and Tao about the nonexistence of Fourier spectra for smooth convex bodies.

math.CA↗
arXiv2003

On pointwise estimates of positive definite functions with given support

The following problem originated from a question due to Paul Turan. Suppose $Ω$ is a convex body in Euclidean space $\RR^d$ or in $\TT^d$, which is symmetric about the origin. Over all positive definite functions supported in $Ω$, and with normalized value 1 at the origin, what is the largest possible value of their integral? From this Arestov, Berdysheva and Berens arrived to pose the analogous pointwise extremal problem for intervals in $\RR$. That is, under the same conditions and normalizations, and for any particular point $z\inΩ$, the supremum of possible function values at $z$ is to be found. However, it turns out that the problem for the real line has already been solved by Boas and Kac, who gave several proofs and also mentioned possible extensions to $\RR^d$ and non-convex domains as well. We present another approach to the problem, giving the solution in $\RR^d$ and for several cases in $\TT^d$. In fact, we elaborate on the fact that the problem is essentially one-dimensional, and investigate non-convex open domains as well. We show that the extremal problems are equivalent to more familiar ones over trigonometric polynomials, and thus find the extremal values for a few cases. An analysis of the relation of the problem for the space $\RR^d$ to that for the torus $\TT^d$ is given, showing that the former case is just the limiting case of the latter. Thus the hiearachy of difficulty is established, so that trigonometric polynomial extremal problems gain recognition again.

math.CA↗
arXiv2002

Translational tilings of the integers with long periods

Suppose that A is a finite set of integers of diameter D. Suppose also that the set of integers B is such that A+B is a tiling of the integers, that is each integer is uniquely expressible as a+b, with a in A, b in B. It is well known that B must be a periodic set in this case. Here we study the relationship between the diameter D of A and the least period T of B. We show that T is at most C exp(C \sqrt D \log D \sqrt{\log\log D}) and that we can have T at least quadratic in D.

math.CO↗
arXiv2002

Tiling and spectral properties of near-cubic domains

We prove that is a measurable domain tiles R or R^2 by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1, and give an example showing that there is no analogue of the tiling result in dimensions 3 and higher.

math.CA↗
arXiv2002

On a problem of Turan about positive definite functions

We study the following question posed by Turan. Suppose K is a convex body in Euclidean space which is symmetric with respect to the origin. Of all positive definite functions supported in K, and with value 1 at the origin, which one has the largest integral? It is probably the case that the extremal function is the indicator of the half-body convolved with itself and properly scaled, but this has been proved only for a small class of domains so far. We add to this class of known "Turan domains" the class of all spectral convex domains. These are all convex domains which have an orthogonal basis of complex exponentials. As a corollary we obtain that all convex domains which tile space by translation are Turan domains. We also give a new proof that the Euclidean ball is a Turan domain.

math.CA↗
arXiv2001

A class of non-convex polytopes that admit no orthonormal basis of exponentials

A conjecture of Fuglede states that a bounded measurable set D, of measure 1, can tile space by translations if and only if the Hilbert space L^2(D) has an orthonormal basis consisting of exponentials exp(i 2 pi lambda x). If D has the latter property it is called spectral. Let D be a polytope with the following property: there is a direction u such that, of all the polytope faces perpendicular to u, the total area of the faces pointing in the positive u direction is more than the total area of the faces pointing in the negative u direction. It is almost obvious that such a polytope D cannot tile space by translation. We prove in this paper that such a domain is also not spectral, which agrees with Fuglede's conjecture.

math.CA↗
arXiv2000

The Steinhaus tiling problem and the range of certain quadratic forms

We give a short proof of the fact that there are no measurable subsets of Euclidean space (in dimension d > 2), which, no matter how translated and rotated, always contain exactly one integer lattice point. In dimension d=2 (the original Steinhaus problem) the question remains open.

math.CA↗
arXiv1999

Non-symmetric convex domains have no basis of exponentials

A conjecture of Fuglede states that a bounded measurable set $Ω$ in space, of measure 1, can tile space by translations if and only if the Hilbert space $L^2(Ω)$ has an orthonormal basis consisting of exponentials. If $Ω$ has the latter property it is called spectral. We generalize a result of Fuglede, that a triangle in the plane is not spectral, proving that every non-symmetric convex domain is not spectral.

math.CA↗
arXiv1999

Packing, tiling, orthogonality and completeness

Let $Ω\subseteq {\bf R}^d$ be an open set of measure 1. An open set $D \subseteq {\bf R}^d$ is called a ``tight orthogonal packing region'' for $Ω$ if $D-D$ does not intersect the zeros of the Fourier Transform of the indicator function of $Ω$ and $D$ has measure 1. Suppose that $Λ$ is a discrete subset of ${\bf R}^d$. The main contribution of this paper is a new way of proving the following result (proved by different methods by Lagarias, Reeds and Wang and, in the case of $Ω$ being the cube, by Iosevich and Pedersen: $D$ tiles ${\bf R}^d$ when translated at the locations $Λ$ if and only if the set of exponentials $E_Λ= \{\exp 2πi λ\cdot x: λ\inΛ\}$ is an orthonormal basis for $L^2(Ω)$. (When $Ω$ is the unit cube in ${\bf R}^d$ then it is a tight orthogonal packing region of itself.) In our approach orthogonality of $E_Λ$ is viewed as a statement about ``packing'' ${\bf R}^d$ with translates of a certain nonnegative function and, additionally, we have completeness of $E_Λ$ in $L^2(Ω)$ if and only if the above-mentioned packing is in fact a tiling. We then formulate the tiling condition in Fourier Analytic language and use this to prove our result.

math.CA↗
arXiv1999

A remark on perturbations of sine and cosine sums

Consider a collection $λ_1<...<λ_N$ of distinct positive integers and the quantities $$ M_1 = M_1(λ_1,...,λ_N) = \max_{0\le x \le 2π} |\sum_{j=1}^N \sin{λ_j x}| $$ and $$ M_2 = M_2(λ_1,...,λ_N) = - \min_{0\le x \le 2π} \sum_{j=1} \cos{λ_j x}. $$ Prompted by a discussion with G. Benke we prove that collections of frequencies $λ_j$ which have $M_1 = o(N)$ or $M_2 = o(N)$ are unstable, in the sense that one can perturb the $λ_j$ by one each and get $M_1 \ge c N$ and $M_2 \ge c N$.

math.CA↗
arXiv1999

On the structure of multiple translational tilings by polygonal regions

We consider polygons with the following ``pairing property'': for each edge of the polygon there is precisely one other edge parallel to it. We study the problem of when such a polygon $K$ tiles the plane multiply when translated at the locations $Λ$, where $Λ$ is a multiset in the plane. The pairing property of $K$ makes this question particularly amenable to Fourier Analysis. After establishing a necessary and sufficient condition for $K$ to tile with a given lattice $Λ$ (which was first found by Bolle for the case of convex polygons-notice that all convex polygons that tile, necessarily have the pairing property and, therefore, our theorems apply to them) we move on to prove that a large class of such polygons tiles only quasi-periodically, which for us means that $Λ$ must be a finite union of translated 2-dimensional lattices in the plane. For the particular case of convex polygons we show that all convex polygons which are not parallelograms tile necessarily quasi-periodically, if at all.

math.MG↗
arXiv1998

On the uniform distribution in residue classes of dense sets of integers with distinct sums

A set ${\cal A} \subseteq \Set{1,...,N}$ is of type $B_2$ if all sums $a+b$, with $a\ge b$, $a,b\in {\cal A}$, are distinct. It is well known that the largest such set is of size asymptotic to $N^{1/2}$. For a $B_2$ set ${\cal A}$ of this size we show that, under mild assumptions on the size of the modulus $m$ and on the difference $N^{1/2}-\Abs{\cal A}$ (these quantities should not be too large) the elements of ${\cal A}$ are uniformly distributed in the residue classes mod $m$. Quantitative estimates on how uniform the distribution is are also provided. This generalizes recent results of Lindström whose approach was combinatorial. Our main tool is an upper bound on the minimum of a cosine sum of $k$ terms, $\sum_1^k \cos{λ_j x}$, all of whose positive integer frequencies $λ_j$ are at most $(2-ε)k$ in size.

math.NT↗