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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.

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., Comment: 13 pages, no figures

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.

By constructing suitable nonnegative exponential sums we give upper bounds on the cardinality of any set $B_q$ in cyclic groups $\ZZ_q$ such that the difference set $B_q-B_q$ avoids cubic residues modulo $q$., Comment: 8 pages

We prove that the fractional chromatic number $\chi_f(\mathbb{R}^2)$ of the unit distance graph of the Euclidean plane is greater than or equal to $4$. Notably, however, we cannot present a finite subgraph $G$ of the plane such that $\chi_f(G)\ge 4$. Instead, we utilize the concept of geometric fractional chromatic number $\chi_{gf}(G)$, which was introduced recently in connection with density bounds for 1-avoiding sets. First, as $G$ ranges over finite subgraphs of the plane, we establish that the supremum of $\chi_f(G)$ is the same as that of $\chi_{gf}(G)$. The proof exploits the amenability of the group of Euclidean transformations in dimension 2 and, as such, we do not know whether the analogous statement holds in higher dimensions. We then present a specific planar unit distance graph $G$ on 27 vertices such that $\chi_{gf}(G)=4$.

We discuss the relation of tiling, weak tiling and spectral sets in finite abelian groups. In particular, in elementary $p$-groups $(\mathbb{Z}_p)^d$, we introduce an averaging procedure that leads to a natural object of study: a 4-tuple of functions which can be regarded as a common generalization of tiles and spectral sets. We characterize such 4-tuples for $d=1, 2$, and prove some partial results for $d=3$.

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ admits an orthogonal basis of exponential functions. Fuglede (1974) conjectured that $\Omega$ 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.

By improving upon previous estimates on a problem posed by L. Moser, we prove a conjecture of Erd\H{o}s that the density of any measurable planar set avoiding unit distances cannot exceed $1/4$. Our argument implies the upper bound of $0.2470$., Comment: 24 pages, 6 figures. Final version, to appear in Mathematical Programming

Suppose that for some unit vectors $b_1,\ldots b_n$ in $\mathbb C^d$ we have that for any $j\neq k$ $b_j$ is either orthogonal to $b_k$ or $|\langle b_j,b_k\rangle|^2 = 1/d$ (i.e. $b_j$ and $b_k$ are unbiased). We prove that if $n=d(d+1)$, then these vectors necessarily form a complete system of mutually unbiased bases, that is, they can be arranged into $d+1$ orthonormal bases, all being mutually unbiased with respect to each other.

A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. A conjecture due to Fuglede (1974) stated that $\Omega$ 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 $\Omega \subset \mathbb{R}^d$ 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 $\Omega$ is a convex polytope. In higher dimensions, this direction of the conjecture remained completely open (even in the case when $\Omega$ 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 $\Omega \subset \mathbb{R}^d$ is a spectral set then $\Omega$ 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 $\Omega \subset \mathbb{R}^d$ to be spectral.

For a set $A$ of points in the plane, not all collinear, we denote by ${\rm tr}(A)$ the number of triangles in any triangulation of $A$; that is, ${\rm tr}(A) = 2i+b-2$ where $b$ and $i$ are the numbers of points of $A$ in the boundary and the interior of $[A]$ (we use $[A]$ to denote "convex hull of $A$"). We conjecture the following analogue of the Brunn-Minkowski inequality: for any two point sets $A,B \subset {\mathbb R}^2$ one has \[ {\rm tr}(A+B)^{\frac12}\geq {\rm tr}(A)^{\frac12}+{\rm tr}(B)^{\frac12}. \] We prove this conjecture in several cases: if $[A]=[B]$, if $B=A\cup\{b\}$, if $|B|=3$, or if none of $A$ or $B$ has interior points., Comment: 30 pages

We improve the best known upper bound on the density of a planar measurable set A containing no two points at unit distance to 0.25442. We use a combination of Fourier analytic and linear programming methods to obtain the result. The estimate is achieved by means of obtaining new linear constraints on the autocorrelation function of A utilizing triple-order correlations in A, a concept that has not been previously studied., Comment: 11 pages, 2 figures

We apply an improvement of the Delsarte LP-bound to give a new proof of the non-existence of finite projective planes of order 6, and uniqueness of finite projective planes of order 7. The proof is computer aided, and it is also feasible to apply to higher orders like 8, 9 and, with further improvements, possibly 10 and 12., Comment: 11 pages

Recently, the authors of the present work (together with M. N. Kolountzakis) introduced a new version of the non-commutative Delsarte scheme and applied it to the problem of mutually unbiased bases. Here we use this method to investigate the existence of a finite projective plane of a given order d. In particular, a short new proof is obtained for the nonexistence of a projective plane of order 6. For higher orders like 10 and 12, the method is non decisive but could turn out to give important supplementary informations.

We present a new approach to the problem of mutually unbiased bases (MUBs), based on positive definite functions on the unitary group. The method provides a new proof of the fact that there are at most $d+1$ MUBs in ${\mathbb C}^d$. It may also lead to a proof of non-existence of complete systems of MUBs in dimension 6 via a conjectured algebraic identity.

In this note we investigate the existence of flat orthogonal matrices, i.e. real orthogonal matrices with all entries having absolute value close to $\frac{1}{\sqrt{n}}$. Entries of $\pm \frac{1}{\sqrt{n}}$ correspond to Hadamard matrices, so the question of existence of flat orthogonal matrices can be viewed as a relaxation of the Hadamard problem., Comment: 10 pages

A $1$-avoiding set is a subset of $\mathbb{R}^n$ that does not contain pairs of points at distance $1$. Let $m_1(\mathbb{R}^n)$ denote the maximum fraction of $\mathbb{R}^n$ that can be covered by a measurable $1$-avoiding set. We prove two results. First, we show that any $1$-avoiding set in $\mathbb{R}^n$ ($n\ge 2$) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than $1$ and points from distinct blocks lie farther than $1$ unit of distance apart from each other) has density strictly less than $1/2^n$. For the special case of sets with block structure this proves a conjecture of Erd\H{o}s asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$., Comment: 16 pages, 1 figure. Contains a Sage script called dstverify.sage, to verify the application of Theorem 3.3. Download the article source to get the script

For infinitely many primes $p=4k+1$ we give a slightly improved upper bound for the maximal cardinality of a set $B\subset \ZZ_p$ such that the difference set $B-B$ contains only quadratic residues. Namely, instead of the "trivial" bound $|B|\leq \sqrt{p}$ we prove $|B|\leq \sqrt{p}-1$, under suitable conditions on $p$. The new bound is valid for approximately three quarters of the primes $p=4k+1$., Comment: 6 pages

We describe general connections between intersective properties of sets in Abelian groups and positive exponential sums. In particular, given a set $A$ the maximal size of a set whose difference set avoids $A$ will be related to positive exponential sums using frequencies from $A$., Comment: 21 pages, a few remarks added, and one theorem slightly extended

We use combinatorial and Fourier analytic arguments to prove various non-existence results on systems of real and complex unbiased Hadamard matrices. In particular, we prove that a complete system of complex mutually unbiased Hadamard matrices (MUHs) in any dimension $d$ cannot contain more than one real Hadamard matrix. We also give new proofs of several known structural results in low dimensions, for $d\le 6$., Comment: 13 pages

We outline a discretization approach to determine the maximal number of mutually unbiased bases in dimension 6. We describe the basic ideas and introduce the most important definitions to tackle this famous open problem which has been open for the last 10 years. Some preliminary results are also listed., Comment: 13 pages

This short note gives an upper bound on the measure of sets $A\subset [0,1]$ such that $x+y=3z$ has no solutions in $A$., Comment: 5 pages

This is a survey about tiling by translation only and related questions and methods, especially those that have to do with Fourier Analysis.

We give an entirely new approach to the problem of mutually unbiased bases (MUBs), based on a Fourier analytic technique in additive combinatorics. The method provides a short and elegant generalization of the fact that there are at most $d+1$ MUBs in $\Co^d$. It may also yield a proof that no complete system of MUBs exists in some composite dimensions -- a long standing open problem., Comment: 11 pages

We give a slight improvement of the best known lower bound for the supremum of autoconvolutions of nonnegative functions supported in a compact interval. Also, by means of explicit examples we disprove a long standing natural conjecture of Schinzel and Schmidt concerning the extremal function for such autoconvolutions., Comment: 15 pages. Some typos have been corrected, the best examples have been slightly improved, two plots of them have been included and some remarks have been added

In this paper we study algorithms for tiling problems. We show that the conditions $(T1)$ and $(T2)$ of Coven and Meyerowitz, conjectured to be necessary and sufficient for a finite set $A$ to tile the integers, can be checked in time polynomial in ${diam}(A)$. We also give heuristic algorithms to find all non-periodic tilings of a cyclic group $Z_N$. In particular we carry out a full classification of all non-periodic tilings of $Z_{144}$., Comment: 13 pages, 1 figure

The aim of this paper is to prove a general version of Pl\"unnecke's inequality. Namely, assume that for finite sets $A$, $B_1, ... B_k$ we have information on the size of the sumsets $A+B_{i_1}+... +B_{i_l}$ for all choices of indices $i_1, ... i_l.$ Then we prove the existence of a non-empty subset $X$ of $A$ such that we have `good control' over the size of the sumset $X+B_1+... +B_k$. As an application of this result we generalize an inequality of \cite{gymr} concerning the submultiplicativity of cardinalities of sumsets., Comment: 8 pages

We extend Freiman's inequality on the cardinality of the sumset of a $d$ dimensional set. We consider different sets related by an inclusion of their convex hull, and one of them added possibly several times., Comment: 6 pages

The aim of this paper is to investigate the cone of non-negative, radial, positive-definite functions in the set of continuous functions on $\R^d$. Elements of this cone admit a Choquet integral representation in terms of the extremals. The main feature of this article is to characterize some large classes of such extremals. In particular, we show that there many other extremals than the gaussians, thus disproving a conjecture of G. Choquet and that no reasonable conjecture can be made on the full set of extremals. The last feature of this article is to show that many characterizations of positive definite functions available in the literature are actually particular cases of the Choquet integral representations we obtain.

For finite sets of integers $A_1, A_2 ... A_n$ we study the cardinality of the $n$-fold sumset $A_1+... +A_n$ compared to those of $n-1$-fold sumsets $A_1+... +A_{i-1}+A_{i+1}+... A_n$. We prove a superadditivity and a submultiplicativity property for these quantities. We also examine the case when the addition of elements is restricted to an addition graph between the sets., Comment: 9 pages

Complex Hadamard matrices have received considerable attention in the past few years due to their appearance in quantum information theory. While a complete characterization is currently available only up to order 5 (in \cite{haagerup}), several new constructions of higher order matrices have appeared recently \cite{dita, karol, BN, MM, sz}. In particular, the classification of {\it self-adjoint} complex Hadamard matrices of order 6 was completed by Beuachamp and Nicoara in \cite{BN}, providing a previously unknown non-affine one-parameter orbit. In this paper we classify all \emph{dephased, symmetric} complex Hadamard matrices \emph{with real diagonal} of order 6. Furthermore, relaxing the condition on the diagonal entries we obtain a new non-affine one-parameter orbit connecting the Fourier matrix $F_6$ and Di\c{t}\u{a}'s matrix $D_6$. This answers a recent question of Bengtsson & al. in \cite{BB}., Comment: 14 pages

We observe that successive applications of known results from the theory of positive systems lead to an {\it efficient general algorithm} for positive realizations of transfer functions. We give two examples to illustrate the algorithm, one of which complements an earlier result of \cite{large}. Finally, we improve a lower-bound of \cite{mn2} to indicate that the algorithm is indeed efficient in general.

Recent methods developed by Tao \cite{tao}, Kolountzakis and Matolcsi \cite{nspec} have led to counterexamples to Fugelde's Spectral Set Conjecture in both directions. Namely, in $\RR^5$ Tao produced a spectral set which is not a tile, while Kolountzakis and Matolcsi showed an example of a non-spectral tile. In search of lower dimensional non-spectral tiles we were led to investigate the Universal Spectrum Conjecture (USC) of Lagarias and Wang \cite{lagwang}. In particular, we prove here that the USC and the ``tile $\to$ spectral'' direction of Fuglede's conjecture are equivalent in any dimensions. Also, we show by an example that the sufficient condition of Lagarias and Szab\'o \cite{lagszab} for the existence of universal spectra is not necessary. This fact causes considerable difficulties in producing lower dimensional examples of tiles which have no spectra. We overcome these difficulties by invoking some ideas of R\'ev\'esz and Farkas \cite{revfark}, and obtain non-spectral tiles in $\RR^3$. Fuglede's conjecture and the Universal Spectrum Conjecture remains open in 1 and 2 dimensions. The 1 dimensional case is closely related to a number theoretical conjecture on tilings by Coven and Meyerowitz \cite{covmey}., Comment: 12 pages

A new construction for the form sum of positive, selfadjoint operators is given in this paper. The situation is a bit more general, because our aim is to add positive, symmetric operators. With the help of the used method, some commutation properties of the form sum extension are observed., Comment: 14 pages

The first representation theorem establishes a correspondence between positive, self-adjoint operators and closed, positive forms on Hilbert spaces. The aim of this paper is to show that some of the results remain true if the underlying space is a reflexive Banach space. In particular, the construction of the Friedrichs extension and the form sum of positive operators can be carried over to this case., Comment: 13 pages

The present paper deals with lower bounds for the norm of products of linear forms. It has been proved by J. Arias-de-Reyna \cite{ARIAS}, that %for ${\mathbb C}^n$, the so-called $n^{\rm th}$ linear polarization constant $c_n({\mathbb C}^n)$ is $n^{n/2}$, for arbitrary $n\in\NN$. The same value for $c_n({\mathbb R}^n)$ is only conjectured. In a recent work A. Pappas and S. R{\'e}v{\'e}sz prove that $c_n({\mathbb R}^n)=n^{n/2}$ for $n \le 5$. Moreover, they show that if the linear forms are given as $f_j(x)=< x,a_j>$, for some unit vectors $a_j$ $(1\leq j\leq n)$, then the product of the $f_j$'s attains at least the value $n^{-n/2}$ at the normalized signed sum of the vectors $\{a_j\}_{j=1}^{n}$ having maximal length. Thus they asked whether this phenomenon remains true for arbitrary $n\in{\mathbb N}$. We show that for vector systems $\{a_j\}_{j=1}^{n}$ close to an orthonormal system, the Pappas-R{\'e}v{\'e}sz estimate does hold true. Furthermore, among these vector systems the only system giving $n^{-n/2}$ as the norm of the product is the orthonormal system. On the other hand, for arbitrary vector systems we answer the question of A. Pappas and S. R{\'e}v{\'e}sz in the negative when $n\in {\mathbb N}$ is large enough. We also discuss various further examples and counterexamples that may be instructive for further research towards the determination of $c_n(\RR^n)$., Comment: 10 pages

In this note we give an example of a set $\W\subset \R^4$ such that $L^2(\W)$ admits an orthonormal basis of exponentials $\{\frac{1}{|\W |^{1/2}}e^{2\pi i x, \xi}\}_{\xi\in\L}$ for some set $\L\subset\R^4$, but which does not tile $\R^4$ by translations. This improves Tao's recent 5-dimensional example, and shows that one direction of Fuglede's conjecture fails already in dimension 4. Some common properties of translational tiles and spectral sets are also proved., Comment: 6 pages

The open problem of determining the exact value of the $n$-th linear polarization constant $c_n$ of $\R^n$ has received considerable attention over the past few years. This paper makes a contribution to the subject by providing a new lower bound on the value of $\sup_{\|{\bf{y}}\|=1}| {\bf{x}}_1,{\bf{y}} ... {\bf{x}}_n,{\bf{y}} |$, where ${\bf{x}}_1, ... ,{\bf{x}}_n$ are unit vectors in $\R^n$. The new estimate is given in terms of the eigenvalues of the Gram matrix $[ {\bf{x}}_i,{\bf{x}}_j ]$ and improves upon earlier estimates of this kind. However, the intriguing conjecture $c_n=n^{n/2}$ remains open., Comment: 7 pages

The present work contributes to the determination of the $n$-th linear polarization constant $c_n(H)$ of an $n$-dimensional real Hilbert space $H$. We provide some new lower bounds on the value of $\sup_{\|y\|=1}| x_1,y >... x_n,y |$, where $x_1, ..., x_n$ are unit vectors in $H$. In particular, the results improve an earlier estimate of Marcus. However, the intriguing conjecture $c_n(H)=n^{n/2}$ remains open., Comment: 8 pages

The aim of this paper is to give a characterization in Hilbert spaces of the generators of $C_0$-semigroups associated with closed, sectorial forms in terms of the convergence of a generalized Trotter's product formula. In the course of the proof of the main result we also present a similarity result which can be of independent interest: for any unbounded generator $A$ of a $C_0$-semigroup $e^{tA}$ it is possible to introduce an equivalent scalar product on the space, such that $e^{tA}$ becomes non-quasi-contractive with respect to the new scalar product., Comment: 12 pages

A basic result in semigroup theory states that every $C_0$-semigroup is quasi-contractive with respect to some appropriately chosen equivalent norm. This paper contains a counterpart of this well-known fact. Namely, by examining the convergence of the Trotter-type formula $(e^{\frac{t}{n}A}P)^n$ (where $P$ denotes a bounded projection), we prove that whenever the generator $A$ is unbounded it is possible to introduce an equivalent norm on the space with respect to which the semigroup is {\it{not}} quasi-contractive., Comment: 4 pages

The Global Positioning System (GPS) provides an excellent educational example as to how the theory of general relativity is put into practice and becomes part of our everyday life. This paper gives a short and instructive derivation of an important formula used in the GPS, and is aimed at graduate students and general physicists. The theoretical background of the GPS (see \cite{ashby}) uses the Schwarzschild spacetime to deduce the {\it approximate} formula, $ds/dt\approx 1+V-\frac{|\vv|^2}{2}$, for the relation between the proper time rate $s$ of a satellite clock and the coordinate time rate $t$. Here $V$ is the gravitational potential at the position of the satellite and $\vv$ is its velocity (with light-speed being normalized as $c=1$). In this note we give a different derivation of this formula, {\it without using approximations}, to arrive at $ds/dt=\sqrt{1+2V-|\vv|^2 -\frac{2V}{1+2V}(\n\cdot\vv)^2}$, where $\n$ is the normal vector pointing outward from the center of Earth to the satellite. In particular, if the satellite moves along a circular orbit then the formula simplifies to $ds/dt=\sqrt{1+2V-|\vv|^2}$. We emphasize that this derivation is useful mainly for educational purposes, as the approximation above is already satisfactory in practice., Comment: 5 pages, revised, over-over-simplified... Does anyone care that the GPS uses an approximate formula, while a precise one is available in just a few lines??? Physicists don't

Suppose we put an $\epsilon$-disk around each lattice point in the plane, and then we rotate this object around the origin for a set $\Theta$ of angles. When do we cover the whole plane, except for a neighborhood of the origin? This is the problem we study in this paper. It is very easy to see that if $\Theta = [0,2\pi]$ then we do indeed cover. The problem becomes more interesting if we try to achieve covering with a small closed set $\Theta$., Comment: 8 pages