Academic literature on the topic 'Fuglede conjecture'

AbstractThe spectral set conjecture, also known as the Fuglede conjecture, asserts that every bounded spectral set is a tile and vice versa. While this conjecture remains open on ${\mathbb R}^1$, there are many results in the literature that discuss the relations among various forms of the Fuglede conjecture on ${\mathbb Z}_n$, ${\mathbb Z}$ and ${\mathbb R}^1$ and also the seemingly stronger universal tiling (spectrum) conjectures on the respective groups. In this paper, we clarify the equivalences between these statements in dimension one. In addition, we show that if the Fuglede conjecture on ${\mathbb R}^1$ is true, then every spectral set with rational measure must have a rational spectrum. We then investigate the Coven–Meyerowitz property for finite sets of integers, introduced in [1], and we show that if the spectral sets and the tiles in ${\mathbb Z}$ satisfy the Coven–Meyerowitz property, then both sides of the Fuglede conjecture on ${\mathbb R}^1$ are true.

Abstract Conjecture 4 of Voiculescu implies that almost normal operators must satisfy a Fuglede-Putnam theorem, namely [T∗, X] is a Hilbert-Schmidt operator whenever [T, X] is in the same class for an arbitrary operator X. In this note, a partial answer to this question is given, namely when X ∈ 𝓒4, the Fuglede-Putnam theorem holds.

The notion of tree entropy was introduced by the author as a normalized limit of the number of spanning trees in finite graphs, but is defined on random infinite rooted graphs. We give some new expressions for tree entropy; one uses Fuglede–Kadison determinants, while another uses effective resistance. We use the latter to prove that tree entropy respects stochastic domination. We also prove that tree entropy is non-negative in the unweighted case, a special case of which establishes Lück's Determinant Conjecture for Cayley-graph Laplacians. We use techniques from the theory of operators affiliated to von Neumann algebras.

The study of Fuglede’s conjecture on the direct product of elementary abelian groups was initiated by Iosevich et al. For the product of two elementary abelian groups the conjecture holds. For Z p 3 \mathbb {Z}_p^3 the problem is still open if p p is prime and p ≥ 11 p\ge 11 . In connection we prove that Fuglede’s conjecture holds on Z p 2 × Z q \mathbb {Z}_{p}^2\times \mathbb {Z}_q by developing a method based on ideas from discrete geometry.

Dans cette thèse, nous résolvons la conjecture de Fuglede sur le corps des nombres p-adiques, et étudions certaines propriétés aléatoires des suites liées à la conjecture de Sarnak, ainsi que leur propriétés oscillantes. Dans la première partie, nous prouvons d'abord la conjecture de Fuglede pour des ensembles ouverts compacts dans Q_p. Celle-ci indique qu'un ensemble ouvert compact dans Q_p est un ensemble spectral si et seulement s'il pave Q_p par translation. Il est également prouvé qu'un ensemble ouvert compact est un ensemble spectral (ou une tuile) si et seulement s'il est p-homogène. Nous caractérisons les ensembles spectraux dans Z / p^n Z ( p>1 premier, n>0 entier) par la propriété de pavage et aussi par leur homogénéité. Finalement, nous montrons la conjecture de Fuglede dans Q_p sans la restriction d'être ouvert compact en montrant que tout ensemble spectral ou toute tuile doivent être ouvert et compact à un ensemble de mesure nulle près. Dans la seconde partie, nous donnons d'abord plusieurs définitions équivalentes d'une suite oscillante en termes de disjonction de différents systèmes dynamiques sur des tores. Ensuite, nous définissons la propriété de Chowla et la propriété de Sarnak pour des suites numériques prenant des valeurs 0 ou des nombres complexes de module 1. Nous prouvons que la propriété de Chowla implique la propriété de Sarnak. Il est également prouvé que pour Lebesgue presque tout b> 1, la suite (e^{2 pi b^n})_{n in N} partage la propriété de Chowla et est par conséquent orthogonale à tout système dynamique topologique d'entropie nulle. Nous discutons également si les échantillons d'une suite aléatoire donnée ont presque sûrement la propriété de Chowla. Nous construisons certaines suites aléatoires dépendantes ayant presque sûrement la propriété de Chowla
In this thesis, we solve Fuglede's conjecture on the field of p-adic numbers, and study some randomness and the oscillating properties of sequences related to Sarnak's conjecture. In the first part, we first prove Fuglede's conjecture for compact open sets in the field Q_p which states that a compact open set in Q_p is a spectral set if and only if it tiles Q_p by translation. It is also proved that a compact open set is a spectral set (or a tile) if and only if it is p-homogeneous. We characterize spectral sets in Z/p^n Z (p>1 prime, n>0 integer) by tiling property and also by homogeneity. Finally, we prove Fuglede's conjecture in Q_p without the assumption of compact open sets and also show that the spectral sets (or tiles) are the sets which differ by null sets from compact open sets. In the second part, we first give several equivalent definitions of oscillating sequences in terms of their disjointness from different dynamical systems on tori. Then we define Chowla property and Sarnak property for numerical sequences taking values 0 or complex numbers of modulus 1. We prove that Chowla property implies Sarnak property. It is also proved that for Lebesgue almost every b>1, the sequence (e^{2 pi b^n})_{n in N} shares Chowla property and consequently is orthogonal to all topological dynamical systems of zero entropy. We also discuss whether the samples of a given random sequence have almost surely Chowla property. Some dependent random sequences having almost surely Chowla property are constructed

In questa tesi ci occupiamo di Canoni Ritmici a Mosaico, che sono composizioni contrappuntistiche puramente ritmiche. I canoni nella musica hanno una tradizione molto lunga; tra questi emergono i canoni ritmici a mosaico (cioè, canoni tali che, dato un tempo, ad ogni battito suona esattamente una voce). Solo nel secolo scorso, a partire dall'analogo problema della fattorizzazione di gruppi abeliani finiti, sono stati studiati i canoni ritmici a mosaico aperiodici: si tratta di canoni che tassellano un certo intervallo di tempo in cui ciascuna voce (voce interna) suona su una sequenza aperiodica di battiti, e anche la sequenza dei battiti iniziali di ogni voce (voce esterna) è aperiodica. Dal punto di vista musicale, l'articolo fondamentale è stato probabilmente quello in quattro parti scritto da D.T. Vuza tra il 1991 e il 1993, mentre la controparte matematica del problema è stata studiata anche prima, ad esempio, da de Bruijn, Sands, ecc., e successivamente, ad esempio, da Coven e Meyerowitz, Jedrzejewski, Amiot, Andreatta, ecc. Non è stata ancora stabilita una teoria approfondita delle condizioni di esistenza e della struttura dei canoni ritmici a mosaico aperiodici. In questa tesi, cerchiamo di dare un contributo a questo affascinante campo. Nel capitolo 2, presentiamo i canoni ritmici a mosaico da un punto di vista matematico e algebrico, concentrandoci sulla loro rappresentazione polinomiale e riportando i risultati fondamentali noti in letteratura. Nel capitolo 3 ci occupiamo di canoni ritmici aperiodici, cioè di canoni in cui in entrambi i ritmi non vi sono strutture interne ripetute: né il ritmo interno né quello esterno si ottengono come ripetizione di un ritmo più breve. Da un punto di vista matematico, sono i canoni più interessanti in quanto diventano un possibile approccio per risolvere la congettura di Fuglede sui domini spettrali. Se viene fornito uno degli insiemi, diciamo $A$, è noto che il problema di trovare un complementare $B$ non ha, in generale, una soluzione univoca. È molto facile trovare canoni a mosaico in cui almeno uno degli insiemi è periodico, cioè è costruito ripetendo un ritmo più breve. Nel Capitolo 4 ci occupiamo della realizzazione di due algoritmi il cui scopo è trovare il ritmo complementare di un dato ritmo aperiodico in un certo periodo $n$. Per enumerare tutti i canoni a mosaico aperiodici, bisogna l’ostacolo della dimensione combinatoria del dominio che diventa ben presto enorme. I principali contributi all'approccio algoritmico al problema sono il modello ILP (Integer Linear Programming) e la codifica SAT per risolvere il problema dei complementari aperiodici. Utilizzando un moderno solutore SAT, siamo stati quindi in grado di calcolare l'elenco completo dei complementari aperiodici di alcune classi di ritmi di Vuza per periodi n = {180, 420, 900}.
In this thesis, we deal with Tiling Rhythmic Canons, which are purely rhythmic contrapuntal compositions. Canons in music have a very long tradition; among these, a few cases of tiling rhythmic canons (i.e., canons such that, given a fixed tempo, at every beat exactly one voice is playing) have emerged. Only in the last century, stemming from the analogous problem of factorizing finite abelian groups, aperiodic tiling rhythmic canons have been studied: these are canons that tile a certain interval of time in which each voice (inner voice) plays at an aperiodic sequence of beats, and the sequence of starting beats of every voice (outer voice) is also aperiodic. From the musical point of view, the seminal paper was probably the four-part article written by D.T. Vuza between 1991 and 1993, while the mathematical counterpart of the problem was studied also before, e.g., by de Bruijn, Sands, etc., and after, e.g., by Coven and Meyerowitz, Jedrzejewski, Amiot, Andreatta, etc. A thorough theory of the conditions of existence and the structure of aperiodic tiling rhythmic canons has not been established yet. In this thesis, we try to give a contribution to this fascinating field. In Chapter 2, we present tiling rhythmic canons from a mathematical and algebraic point of view, focusing on their polynomial representation and reporting the fundamental results known in the literature. In Chapter 3, we deal with aperiodic rhythmic canons, that is canons in which in both rhythms there are no repeated inner structures: neither the inner nor the outer rhythm is obtained as a repetition of a shorter rhythm. From a mathematical point of view, they are the most interesting canons since they become a possible approach to solving the Fuglede conjecture on spectral domains. If one of the sets, say $A$, is given, it is well-known that the problem of finding a complement $B$ has, in general, no unique solution. It is very easy to find tiling canons in which at least one of the sets is periodic, i.e., it is built by repeating a shorter rhythm. In Chapter 4 we deal with the realization of two algorithms whose purpose is to find the complementary tiling rhythm of a given aperiodic rhythm in a certain period $n$. To enumerate all aperiodic tiling canons, one must overcome the problem that the combinatorial size of the domain becomes very soon enormous. The main contributions to the algorithmic approach to the problem are the Integer Linear Programming (ILP) model and the SAT Encoding to solve the Aperiodic Tiling Complements Problem. Using a modern SAT solver, we have been therefore able to compute the complete list of aperiodic tiling complements of some classes of Vuza rhythms for periods n = {180, 420, 900}.