Related papers: Distribution of gaps between eigenangles of Hecke …

In 1966, H. Widom proved an asymptotic formula for the distribution of eigenvalues of the $N\times N$ truncated Hilbert matrix for large values of $N$. In this paper, we extend this formula to Hankel matrices with symbols in the class of…

We prove that the statistics of the period of the continued fraction expansion of certain sequences of quadratic irrationals from a fixed quadratic field approach the `normal' statistics given by the Gauss-Kuzmin measure. As a by-product,…

Let $(\lambda_f(n))_{n\geqslant1}$ be the Hecke eigenvalues of a holomorphic cusp form $f$. We prove that the exponent of distribution of $\lambda_f*1$ in arithmetic progressions is as large as $\frac{1}{2}+\frac{1}{46}$ when the modulus…

Let $A$ be a central division algebra of prime degree $p$ over $\mathbb{Q}$. We obtain subconvex hybrid bounds, uniform in both the eigenvalue and the discriminant, for the sup-norm of Hecke-Maass forms on the compact quotients of…

The distribution of the ratios of consecutive eigenvalue spacings of random matrices has emerged as an important tool to study spectral properties of many-body systems. This article numerically investigates the eigenvalue ratios…

This paper is concerned with {an extension and reinterpretation} of previous results on the variational characterization of eigenvalues in gaps of the essential spectrum of self-adjoint operators. {We state} two general abstract results on…

We explicitly compute the limiting gap distribution for slopes of saddle connections on the flat surface associated to the regular octagon with opposite sides identified. This is the first such computation where the Veech group of the…

We establish the Sato-Tate equidistribution of Hecke eigenvalues on average for families of Hecke--Maass cusp forms on SL(n,R)/SO(n). For each of the principal, symmetric square and exterior square L-functions we verify that the families…

We consider the empirical eigenvalue distribution of an $m\times m$ principle submatrix of an $n\times n$ random unitary matrix distributed according to Haar measure. Earlier work of Petz and R\'effy identified the limiting spectral measure…

Let A(x) be a holomorphic family of bounded self-adjoint operators on a separable Hilbert space H and let A(x)_n be the orthogonal compressions of A(x) to the span of first n elements of an orthonormal basis of H. The problem considered…

We study the gaps between consecutive prime numbers directly through Eratosthenes sieve. Using elementary methods, we identify a recursive relation for these gaps and for specific sequences of consecutive gaps, known as constellations.…

We study the gaps $\Delta p_n$ between consecutive rank sizes in $r$-differential posets by introducing a projection operator whose matrix entries can be expressed in terms of the number of certain paths in the Hasse diagram. We strengthen…

We characterize the Maass cusp forms for Hecke congruence subgroups of prime level as 1-eigenfunctions of a finite-term transfer operator.

We are interested in two random matrix ensembles related to permutations: the ensemble of permutation matrices following Ewens' distribution of a given parameter $\theta >0$, and its modification where entries equal to $1$ in the matrices…

In this paper we give an overview of recent results on (upper and lower) discrepancy estimates for (concrete) sequences in the unit-cube. In particular we also give an overview of discrepancy estimates for certain classes of hybrid…

In this paper we obtain explicit formulas for the traces of Hecke operators on spaces of cusp forms in certain instances related to arithmetic triangle groups. These expressions are in terms of hypergeometric character sums over finite…

We obtain multidimensional metric uniform distribution results involving sequences in ${\mathbb R}^k$ parametrized by analytic curves. Our theorems extend the classical theorems of Weyl and Koksma in a variety of ways. One of our main…

Given a homogeneous space $X = G/\Gamma$ with $G$ containing the group $H = (\mathrm{SO}(n,1))^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an analytic curve $\phi: I=[a,b] \rightarrow H$, we will show that if $\phi$ satisfies…

The concept of uniform distribution in $[0,1]$ is extended for a certain strictly separated maximal (in the sense of cardinality) family $(\lambda_t)_{t \in [0,1]}$ of invariant extensions of the linear Lebesgue measure $\lambda$ in…

The main contribution of this article is a new prior distribution over directed acyclic graphs, which gives larger weight to sparse graphs. This distribution is intended for structured Bayesian networks, where the structure is given by an…