Fuglede's conjecture on cyclic groups of order $p^nq$, Discrete Analysis 2017:12, 16 pp. A Fuglede from 1974 states that a measurable set $E\subset \mathbb R^n$ positive Lebesgue measure tiles $\mathbb by translations if and only the space $L^2(E)$ admits an orthonormal basis exponential functions $\{ e^{2\pi i \lambda\cdot x}:\ \lambda\in\Lambda\}$. (The $\Lambda$ is called _spectrum_ for $E$.) We now know false in dimensions 3 higher, with counterexamples due to Tao, Kolountzakis,...

The purpose of this paper is to investigate the properties spectral and tiling subsets cyclic groups, with an eye towards set conjecture in one dimension, which states that a bounded measurable subset $\mathbb{R}$ accepts orthogonal basis exponentials if only it tiles by translations. This strongly connected its discrete counterpart, namely every finite group, tile. tools presented herein are refinements recent ones used setting groups; structure vanishing sums roots unity prevalent notion...

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

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups square-free order, initiated by Laba and Meyerowitz, but spectral-tile far from being well-understood. product at most three primes as order group was studied intensely in last couple years. In this paper we study case when four different prove that Fuglede's conjecture holds case.

The theory of Gabor frames functions defined on finite abelian groups was initially developed in order to better understand the properties over reals.However, during last twenty years topic has acquired an interest its own.One fundamental questions asked this setting is existence full spark frames.The author proved [21], as well constructed such frames, when underlying group cyclic.In paper, we resolve non-cyclic case; particular, show that there can be no windows groups.We also prove all...

The purpose of this paper is to establish an inequality connecting the lattice point enumerator a 0-symmetric convex body with its successive minima. To end, we introduce optimization problem whose solution refines former methods, thus producing better upper bound. In particular, show that analogue Minkowski’s second theorem on minima volume replaced by true up exponential factor, base approximately 1.64.

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

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

Tao (2018) showed that in order to prove the Lonely Runner Conjecture (LRC) up $n+1$ runners it suffices consider positive integer velocities of $n^{O(n^2)}$. Using zonotopal reinterpretation conjecture due first and third authors (2017) we here drastically improve this result, showing $\binom{n+1}{2}^{n-1} \le n^{2n}$ are enough. We same finite-checking with bound, for more general \emph{shifted} (sLRC), except case our result depends on solution a question, dub \emph{Lonely Vector Problem}...

Abstract We present an approach to Fuglede’s conjecture in $$\mathbb {Z}_p^3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>Z</mml:mi> <mml:mi>p</mml:mi> <mml:mn>3</mml:mn> </mml:msubsup> </mml:math> using linear programming bounds, obtaining the following partial result: if $$A\subseteq \mathbb <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>⊆</mml:mo> </mml:mrow> with $$p^2-p\sqrt{p}+\sqrt{p}&lt;|A|&lt;p^2$$ <mml:msup> <mml:mn>2</mml:mn> </mml:msup>...

We define certain natural finite sums of $n$'th roots unity, called $G_P(n)$, that are associated to each convex integer polytope $P$, and which generalize the classical $1$-dimensional Gauss sum $G(n)$ defined over $\mathbb Z/ {n \mathbb Z}$, higher dimensional abelian groups polytopes. consider Weyl group $\mathcal{W}$, generated by reflections with respect coordinate hyperplanes, as well all permutations coordinates; further, we let $\mathcal G$ be $\mathcal{W}$ translations in Z^d$....

We are concerned with the computational problem of determining covering radius a rational polytope. This parameter is defined as minimal dilation factor that needed for lattice translates correspondingly dilated polytope to cover whole space. As our main result, we describe new algorithm this problem, which simpler, more efficient and easier implement than only prior Kannan (1992). Motivated by variant famous Lonely Runner Conjecture, use its geometric interpretation in terms radii...

We present a new infinite family of full spark frames in finite dimensions arising from unitary group representation, where the underlying is semi-direct product cyclic by automorphisms. The only previously known algebraically constructed families were harmonic, Gabor and Dihedral frames. Our construction hinges on theorem that requires no structure. Additionally, we illustrate our results providing explicit constructions

The purpose of this paper is to investigate the properties spectral and tiling subsets cyclic groups, with an eye towards set conjecture in one dimension, which states that a bounded measurable subset $\mathbb{R}$ accepts orthogonal basis exponentials if only it tiles by translations. This strongly connected its discrete counterpart, namely every finite group, tile. tools presented herein are refinements recent ones used setting groups; structure vanishing sums roots unity prevalent notion...

The tile-spectral direction of the discrete Fuglede-conjecture is well-known for cyclic groups square-free order, initiated by Laba and Meyerowitz, but spectral-tile far from being well-understood. product at most three primes as order group was studied intensely in last couple years. In this paper we study case when four different prove that Fuglede's conjecture holds case.