Related papers: On the CLT for rotations and BV functions

Let $A_f(1,n)$ be the normalized Fourier coefficients of a Hecke-Maass cusp form $f$ for $SL_3(\mathbb{Z})$ and $$ r_3(n)=\#\left\{(n_1,n_2,n_3)\in \mathbb{Z}^3:n_1^2+n_2^2+n_3^2=n\right\}. $$ Let $1\leq h\leq X$ and $\phi(x)$ be a smooth…

Let $\{X(\mathbf{t}):\mathbf{t}=(t_1, t_2, \ldots, t_d)\in[0,\infty)^d\}$ be a centered stationary Gaussian field with almost surely continuous sample paths, unit variance and correlation function $r$ satisfying conditions $r(\mathbf{t})<1$…

The computation of the amplitude, $\alpha$, of asymptotic standing wave tails of weakly delocalized, stationary solutions in a fifth-order Korteweg-de Vries equation is revisited. Assuming the coefficient of the fifth order derivative term,…

We derive an asymptotic error formula for Gauss--Legendre quadrature applied to functions with limited regularity, using the contour-integral representation of the remainder term. To address the absence of uniformly valid approximations of…

We study limit cycles of nonlinear oscillators described by the equation $\ddot x + \nu F(\dot x) + x =0$. Depending on the nonlinearity this equation may exhibit different number of limit cycles. We show that limit cycles correspond to…

Let $X$ be a $G$-homogeneous space over a number field $k$ such that $X\cong G_\gamma\backslash G$. Here, $G$ is a simply connected semisimple group over $k$ and $\gamma\in G(k)$ whose centralizer $G_\gamma$ is a maximal torus in $G$ which…

Given an irrational vector $\alpha$ in $\mathbb{R}^{d}$, a continuous function $f(x)$ on the torus $\mathbb{T}^{d}$ and suitable weights $\Phi(N,n)$ such that $\sum_{n=-\infty}^{+\infty}\Phi(N,n)=1$, we estimate the speed of convergence to…

We give a two-dimensional central limit theorem (CLT) for the second-order quadratic variation of the centered Gaussian processes on $[0,T]$. Though the approach we use is well known in the literature, the conditions under which the CLT…

We are interested in the asymptotic behavior of orthogonal polynomials of the generalized Jacobi type as their degree $n$ goes to $\infty$. These are defined on the interval $[-1,1]$ with weight function…

In \cite{BAMU}, an ergodic theorem \`a la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the…

Recent work has shown a deep connection between semilocal approximations in density functional theory and the asymptotics of the sum of the WKB semiclassical expansion for the eigenvalues. However, all examples studied to date have…

Isotropic functions of positions $\hat{\bf r}_1, \hat{\bf r}_2,\ldots, \hat{\bf r}_N$, i.e. functions invariant under simultaneous rotations of all the coordinates, are conveniently formed using spherical harmonics and Clebsch-Gordan…

Given a positive increasing function $\varphi$, we show that for a full measure set of vectors $\alpha\in \mathbb{R}^d$, the maximal ergodic discrepancy of the $d$-linear form sequence $\left\{\sum_{1\le i\le d} k_i\alpha_i \mod…

We extend Gaussian perturbation models in classical functional data analysis to the three-dimensional rotational group where a zero-mean Gaussian process in the Lie algebra under the Lie exponential spreads multiplicatively around a central…

We obtain asymptotic for the quantity $\int_0^1 \bigg|\sum_{n\le X}\tau_k(n)e(n\alpha)\bigg|d\alpha$ where $\tau_k(n) = \sum_{d_1\dots d_k = n} 1$. This follows from a quick application of the circle method. Along the way, we find minor arc…

We find the asymptotics of the series $\sum_{n=1}^\infty (-1)^n n^{-1} \exp(-t/n)$ as $t\to+\infty$. The answer is an oscillating function of $t$ dominated by $\exp(-(2\pi t)^{1/2})$. The intermediate step is to find the asymptotics of the…

The ergodic theory and geometry of the Julia set of meromorphic functions on the complex plane with polynomial Schwarzian derivative is investigated under the condition that the forward trajectory of asymptotic values in the Julia set is…

We conduct the multifractal analysis of the level sets of the asymptotic behavior of almost-additive continuous potentials $(\phi_n)_{n=1}^\infty$ on a topologically mixing subshift of finite type $X$ endowed itself with a metric associated…

Rotations on the circle by irrational numbers give rise to uniquely ergodic Sturm dynamical systems. We show that rotations by badly approximable irrationals have the property of fast ergodicity. It was shown recently that any Sturmian…

We show that any $q$-multiplicative sequence which is \emph{oscillating} of order $1$, i.e.\ does not correlate with linear phase functions $e^{2\pi i n\alpha}$ ($\alpha \in \mathbb{R})$, is Gowers uniform of all orders, and hence in…