We show that the spectral set conjecture by Fuglede [6] holds in the setting of cyclic groups of order p n q, where p, q are distinct primes and n ≥ 1. This means that a subset E of such a group G tiles the group by translation (G can be partitioned into translates of E) if and only if there exists an orthogonal basis of L 2 (E) consisting of group characters. The main ingredient of the present proof is the structure of vanishing sums of roots of unity of order N, where N has at most two prime divisors; the extension of this proof to the case of cyclic groups of order p n q m seems therefore feasible. The only previously known infinite family of cyclic groups, for which Fuglede's conjecture is verified in both directions, is that of cyclic p-groups, i.e. Z p n .

We investigate the discrete Fuglede's conjecture and Pompeiu problem on finite abelian groups and develop a strong connection between the two problems. We give a geometric condition under which a multiset of a finite abelian group has the discrete Pompeiu property. Using this description and the revealed connection we prove that Fuglede's conjecture holds for Z p n q 2 , where p and q are different primes. In particular, we show that every spectral subset of Z p n q 2 tiles the group. Further, using our combinatorial methods we give a simple proof for the statement that Fuglede's conjecture holds for Z 2 p .

Abstract. The purpose of this note is to present a proof of the existence of Gabor frames in general linear position in all finite dimensions. The tools developed in this note are also helpful towards an explicit construction of such a frame, which is carried out in the last section. This result has applications in signal recovery through erasure channels, operator identification, and time-frequency analysis.

Abstract. The goal of this paper is twofold; first, show the equivalence between certain problems in geometry, such as view-obstruction and billiard ball motions, with the estimation of covering radii of lattice zonotopes. Second, we will estimate upper bounds of said radii by virtue of the Flatness Theorem. These problems are similar in nature with the famous lonely runner conjecture.

Abstract. This note concerns the so-called pyjama problem, whether it is possible to cover the plane by finitely many rotations of vertical strips of half-width ε. We first prove that there exist no periodic coverings for ε < 1 3 . Then we describe an explicit (non-periodic) construction for ε = 1 3 − 1 48 . Finally, we use a compactness argument combined with some ideas from additive combinatorics to show that a finite covering exists for ε = 1 5 . The question whether ε can be arbitrarily small remains open.

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The main result of this paper is an inequality relating the lattice point enumerator of a 3-dimensional, 0-symmetric convex body and its successive minima. This is an example of generalization of Minkowski's theorems on successive minima, where the volume is replaced by the discrete analogue, the lattice point enumerator. This problem is still open in higher dimensions, however, we introduce a stronger conjecture that shows a possibility of proof by induction on the dimension.

Abstract. The theory of Gabor frames of functions defined on finite abelian groups was initially developed in order to better understand the properties of Gabor frames of functions defined over the reals. However, during the last twenty years the topic has acquired an interest of its own. One of the fundamental questions asked in this finite setting is the existence of full spark Gabor frames. The author proved the existence [21], as well as constructed such frames, when the underlying group is finite cyclic. In this paper, we resolve the non-cyclic case; in particular, we show that there can be no full spark Gabor frames of windows defined on finite abelian non-cyclic groups. We also prove that all eigenvectors of certain unitary matrices in the Clifford group in odd dimensions generate spark deficient Gabor frames. Finally, similarities between the uncertainty principles concerning the finite dimensional Fourier transform and the short-time Fourier transform are discussed. IntroductionThe Gabor frame of a function f ∈ L 2 (R) is the set of all time-frequency translates of f , that is, the set of all functions of the form e 2πixy f (x−t), for y, t ∈ R, and it is a fundamental concept in time-frequency analysis and frame theory [25]. The function f usually represents a signal, t the time delay, and the pointwise multiplication by e 2πixy is the frequency "shift". Through sampling and periodization [8] one passes to the finite version of a Gabor frame, namely the shift-frequency translates of a complex function defined on a finite cyclic group. Even though finite dimensional Gabor frames were studied in order to analyze the properties of continuous signals, they later developed an interest of their own.Up to multiplication by roots of unity, a finite dimensional Gabor frame is the same as a Weyl-Heisenberg orbit, and this terminology is much more prevalent in mathematical physics and quantum information theory. A conjecture by Zauner [33] states that for every dimension N there are vectors (called "fiducials") whose WH orbit is equiangular. This means that the expression | u, v | is constant for every pair of distinct vectors u, v within this orbit. This is also known as the SIC-POVM problem which has attracted a lot of attention lately due to the vast connections to scientific areas such as quantum cryptography [26], quantum tomography [27], and algebraic number theory, especially Hilbert's 12th problem for real quadratic fields [2,3,4,5]. Such a WH orbit would then produce the maximal possible number of vectors in C N that are pairwise equiangular, namely N 2 [29]. Yet another terminology that appears for this phenomenon is maximal equiangular tight frame (or maximal ETF for short) [11], which is a special case of the packing problem in the setting of projective spaces. The interest of the algebraic construction of families of ETFs has also increased due to applications to signal processing [12,16,17].A conjecture by Heil, Ramanathan, and Topiwala from 1996 [14] states that any finite set of a Gabor frame of a ...