arXiv subjects
Publications and source records attributed to Máté Matolcsi.
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.
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 Erdős asserting that $m_1(\mathbb{R}^2) < 1/4$. Second, we use linear programming and harmonic analysis to show that $m_1(\mathbb{R}^2) \leq 0.258795$.
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 Diţă's matrix $D_6$. This answers a recent question of Bengtsson & al. in \cite{BB}.
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{é}v{é}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{é}v{é}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{é}v{é}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 $c_n(\RR^n)$.
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 Szabó \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 Révész 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}.
Applications in quantum information theory and quantum tomography have raised current interest in complex Hadamard matrices. In this note we investigate the connection between tiling Abelian groups and constructions of complex Hadamard matrices. First, we recover a recent very general construction of complex Hadamard matrices due to Dita via a natural tiling construction. Then we find some necessary conditions for any given complex Hadamard matrix to be equivalent to a Dita-type matrix. Finally, using another tiling construction, due to Szabo, we arrive at new parametric families of complex Hadamard matrices of order 8, 12 and 16, and we use our necessary conditions to prove that these families do not arise with Dita's construction. These new families complement the recent catalogue of complex Hadamard matrices of small order.