Weyl’s equidistribution theorem

Associated to any finite simple graph \Gamma is the chromatic polynomial \mathcal{P}_\Gamma(q) whose complex zeros are called the chromatic zeros of \Gamma . A hierarchical lattice is a sequence of finite simple graphs \{\Gamma_n\}_{n=0}^\infty built recursively using a substitution rule expressed in terms of a generating graph. For each n , let \mu_n denote the probability measure that assigns a Dirac measure to each chromatic zero of \Gamma_n . Under a mild hypothesis on the generating graph, we prove that the sequence \mu_n converges to some measure \mu as n tends to infinity. We call \mu the limiting measure of chromatic zeros associated to \{\Gamma_n\}_{n=0}^\infty . In the case of the diamond hierarchical lattice we prove that the support of \mu has Hausdorff dimension two. The main techniques used come from holomorphic dynamics and more specifically the theories of activity/bifurcation currents and arithmetic dynamics. We prove a new equidistribution theorem that can be used to relate the chromatic zeros of a hierarchical lattice to the activity current of a particular marked point. We expect that this equidistribution theorem will have several other applications.

The present paper is motivated by the observation that Weyl's equidistribution theorem for real sequences on a bounded interval can be formulated in a way which is also meaningful for sequences of selfadjoint operators on a Hilbert space. We shall provide general results on weak convergence of operator measures which yield this version of Weyl's theorem as a corollary. Further, by combining the above results with the von Neumann we will obtain a Cesaro convergence property, equivalently, an ergodic theorem, which is valid for all (projectionvalued) spectral measures whose support is in a bounded interval, as well as for the more general class of positive operator-valued measures. Within the same circle of ideas we deduce a convergence property which completely characterizes those spectral measures associated with strongly mixing unitary transformations. The final sections are devoted to applications of the preceding results in the study of complexvalued Borel measures as well as to an extension of our results to summability methods other than CesAro convergence. In particular, we obtain a complete characterization, in purely measure theoretic terms, of those complex measures on a bounded interval whose Fourier-Stieltjes coefficients converge to zero.

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We address the problems of extracting information generated by one dimensional periodic point processes. These problems arise in numerous situations, from astronomy and biomedical applications to reliability and quality control and signal processing. We divide our analysis into two cases, namely single and then multiple source(s). We wish to extract the fundamental period of the generator(s), and, in the second case, to deinterleave the processes. We present two algorithms, designed to work on all one dimensional periodic processes, but in particular on sparse datasets where other procedures break down. The first algorithm works on data from single period processes, computing an estimate of the underlying period. It is extremely computationally efficient and straightforward, and works on all single period processes, but in particular on sparse datasets where others break down. Its justification, however, rests on some deep mathematics, including a probabilistic interpretation of the Riemann zeta function. We then build upon this procedure to analyze data from multiple periodic processes. This second procedure relies on the Riemann zeta function, Weyl's equidistribution theorem, and Wiener's periodogram.

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Weyl's equidistribution theorem

Weyl's equidistribution theorem

We construct interval exchange transformations on four intervals satisfying a strong irrationality condition and having exactly two ergodic invariant probability measures. This shows that although Kronecker’s theorem remains true for interval exchange transformations, the Weyl equidistribution theorem is false even under the strongest irrationality assumptions.

We give a survey of some aspects of the Riemann Hypothesis over finite fields, as it was proved by Deligne, and its applications to analytic number theory. In particular, we concentrate on the formalism leading to Deligne’s Equidistribution Theorem.

We prove an equidistribution theorem à la Bader–Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(–1) spaces thanks to an equidistribution theorem of T. Roblin. This result can be viewed as a von Neumann’s mean ergodic theorem for quasi-invariant measures. In particular, this approach gives a dynamical proof of the fact that boundary representations are irreducible. Moreover, we prove some equidistribution results for conformal densities using elementary techniques from harmonic analysis.

A classical result of MacMahon states that inversion number and major index have the same distribution over permutations of a given multiset. In this work we prove a strengthening of this theorem originally conjectured by Haglund. Our result can be seen as an equidistribution theorem over the ordered partitions of a multiset into sets, which we call ordered multiset partitions. Our proof is bijective and involves a new generalization of Carlitz's insertion method. As an application, we develop refined Macdonald polynomials for hook shapes. We show that these polynomials are symmetric and give their Schur expansion. Un résultat classique de MacMahon affirme que nombre d’inversion et l’indice majeur ont la même distribution sur permutations d’un multi-ensemble donné. Dans ce travail, nous démontrons un renforcement de ce théorème origine conjecturé par Haglund. Notre résultat peut être considéré comme un théorème d’équirépartition sur les partitions ordonnées d’un multi-ensemble en ensembles, que nous appellerons partitions de multiset commandés. Notre preuve est bijective et implique une nouvelle généralisation de la méthode d’insertion de Carlitz. Comme application, nous développons des polynômes de Macdonald raffinés pour formes d’hameçons. Nous montrons que ces polynômes sont symétriques et donnent leur expansion Schur.

Previous studies of kinetic transport in the Lorentz gas have been limited to cases where the scatterers are distributed at random (e.g. at the points of a spatial Poisson process) or at the vertices of a Euclidean lattice. In the present paper we investigate quasicrystalline scatterer configurations, which are non-periodic, yet strongly correlated. A famous example is the vertex set of the Penrose tiling. Our main result proves the existence of a limit distribution of the free path length, which answers a question of Wennberg. The limit distribution is characterised by a certain random variable on the space of higher dimensional lattices, and is distinctly different from the exponential distribution observed for random scatterer configurations. The key ingredients in the proofs are equidistribution theorems on homogeneous spaces, which follow from Ratner's measure classification.

The block number of a permutation is the maximum number of components in its expression as a direct sum. We show that, for 321-avoiding permutations, the set of left-to-right maxima has the same distribution when the block number is assumed to be k, as when the last descent of the inverse is assumed to be at position \(n - k\). This result is analogous to the Foata–Schützenberger equidistribution theorem, and implies that the quasi-symmetric generating function of the descent set over 321-avoiding permutations with a prescribed number of blocks is Schur-positive.

Further extensions of Haglund-Remmel-Wilson identity

We prove an equidistribution theorem of positive closed currents for a certain class of birational maps [Formula: see text] of algebraic degree [Formula: see text] satisfying [Formula: see text], where [Formula: see text] is the inverse of [Formula: see text] and [Formula: see text] are the sets of indeterminacy for [Formula: see text], respectively.