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.