Extended Tauberian Theorem for the weighted mean Method of Summability

We prove a new Tauberian-like theorem that establishes the slow oscillation of a real sequence u = (un) on the basis of the weighted mean summability of its generator sequence and some conditions.Доведено нову теорему тауберового типу, яка встановлює повільні коливання дійсної послідовності u = (u n )) на основі збіжності її генеруючої послідовності у зважених середніх та певних умов

Tauberian conditions for a general limitable method

Let (un) be a sequence of real numbers, L an additive limitable method with some property, and and different spaces of sequences related to each other. We prove that if the classical control modulo of the oscillatory behavior of (un) in is a Tauberian condition for L, then the general control modulo of the oscillatory behavior of integer order m of (un) in or is also a Tauberian condition for L

Dynamics of continued fractions and distribution of modular symbols

We formulate a thermodynamical approach to the study of distribution of modular symbols, motivated by the work of Baladi-Vall\'ee. We introduce the modular partitions of continued fractions and observe that the statistics for modular symbols follow from the behavior of modular partitions. We prove the limit Gaussian distribution and residual equidistribution for modular partitions as a vector-valued random variable on the set of rationals whose denominators are up to a fixed positive integer by studying the spectral properties of transfer operator associated to the underlying dynamics. The approach leads to a few applications. We show an average version of conjectures of Mazur-Rubin on statistics for the period integrals of an elliptic newform. We further observe that the equidistribution of mod $p$ values of modular symbols leads to mod $p$ non-vanishing results for special modular $L$-values twisted by a Dirichlet character.Comment: 42 page

Geodesic bi-angles and Fourier coefficients of restrictions of eigenfunctions

This article concerns joint asymptotics of Fourier coefficients of restrictions of Laplace eigenfunctions $\phi_j$ of a compact Riemannian manifold to a submanifold $H \subset M$. We fix a number $c \in (0,1)$ and study the asymptotics of the thin sums, $$N^{c} _{\epsilon, H }(\lambda): = \sum_{j, \lambda_j \leq \lambda} \sum_{k: |\mu_k - c \lambda_j | < \epsilon} \left| \int_{H} \phi_j \overline{\psi_k}dV_H \right|^2$$ where $\{\lambda_j\}$ are the eigenvalues of $\sqrt{-\Delta}_M,$ and $\{(\mu_k, \psi_k)\}$ are the eigenvalues, resp. eigenfunctions, of $\sqrt{-\Delta}_H$. The inner sums represent the `jumps' of $N^{c} _{\epsilon, H }(\lambda)$ and reflect the geometry of geodesic c-bi-angles with one leg on $H$ and a second leg on $M$ with the same endpoints and compatible initial tangent vectors $\xi \in S^c_H M, \pi_H \xi \in B^* H$, where $\pi_H \xi$ is the orthogonal projection of $\xi$ to $H$. A c-bi-angle occurs when $\frac{|\pi_H \xi|}{|\xi|} = c$. Smoothed sums in $\mu_k$ are also studied, and give sharp estimates on the jumps. The jumps themselves may jump as $\epsilon$ varies, at certain values of $\epsilon$ related to periodicities in the c-bi-angle geometry. Subspheres of spheres and certain subtori of tori illustrate these jumps. The results refine those of the previous article (arXiv:2011.11571) where the inner sums run over $k: | \frac{\mu_k}{\lambda_j} - c| \leq \epsilon$ and where geodesic bi-angles do not play a role.Comment: 51 pages. Referee's comments incorporated. To appear in Pure and Applied Analysi

Asymptotic and exact expansions of heat traces

We study heat traces associated with positive unbounded operators with compact inverses. With the help of the inverse Mellin transform we derive necessary conditions for the existence of a short time asymptotic expansion. The conditions are formulated in terms of the meromorphic extension of the associated spectral zeta-functions and proven to be verified for a large class of operators. We also address the problem of convergence of the obtained asymptotic expansions. General results are illustrated with a number of explicit examples.Comment: 44 LaTeX pages, 2 figure

Euclidean algorithms are Gaussian

This study provides new results about the probabilistic behaviour of a class of Euclidean algorithms: the asymptotic distribution of a whole class of cost-parameters associated to these algorithms is normal. For the cost corresponding to the number of steps Hensley already has proved a Local Limit Theorem; we give a new proof, and extend his result to other euclidean algorithms and to a large class of digit costs, obtaining a faster, optimal, rate of convergence. The paper is based on the dynamical systems methodology, and the main tool is the transfer operator. In particular, we use recent results of Dolgopyat.Comment: fourth revised version - 2 figures - the strict convexity condition used has been clarifie

Spin-Statistics for Black Hole Microstates

The gravitational path integral can be used to compute the number of black hole states for a given energy window, or the free energy in a thermal ensemble. In this article we explain how to use the gravitational path integral to compute the separate number of bosonic and fermionic black hole microstates. We do this by comparing the partition function with and without the insertion of $(-1)^{\sf F}$. In particular we introduce a universal rotating black hole that contributes to the partition function in the presence of $(-1)^{\sf F}$. We study this problem for black holes in asymptotically flat space and in AdS, putting constraints on the high energy spectrum of holographic CFTs (not necessarily supersymmetric). Finally, we analyze wormhole contributions to related quantities.Comment: 34 pages; v2: references adde

Eikonalization of Conformal Blocks

Classical field configurations such as the Coulomb potential and Schwarzschild solution are built from the t-channel exchange of many light degrees of freedom. We study the CFT analog of this phenomenon, which we term the `eikonalization' of conformal blocks. We show that when an operator $T$ appears in the OPE $\mathcal{O}(x) \mathcal{O}(0)$, then the large spin $\ell$ Fock space states $[TT \cdots T]_{\ell}$ also appear in this OPE with a computable coefficient. The sum over the exchange of these Fock space states in an $\langle \mathcal{O} \mathcal{O} \mathcal{O} \mathcal{O} \rangle$ correlator build the classical `$T$ field' in the dual AdS description. In some limits the sum of all Fock space exchanges can be represented as the exponential of a single $T$ exchange in the 4-pt correlator of $\mathcal{O}$. Our results should be useful for systematizing $1/\ell$ perturbation theory in general CFTs and simplifying the computation of large spin OPE coefficients. As examples we obtain the leading $\log \ell$ dependence of Fock space conformal block coefficients, and we directly compute the OPE coefficients of the simplest `triple-trace' operators.Comment: 32+17 pages, 6 figures; references added, discussion of eikonal limit clarifie