Related papers: Equidistribution of Kronecker sequences along clos…

An important result of H. Weyl states that for every sequence $\left(a_{n}\right)_{n \geq 1}$ of distinct positive integers the sequence of fractional parts of $\left(a_{n} \alpha \right)_{n\geq 1}$ is uniformly distributed modulo one for…

In this short note, we prove that the one-dimensional Kronecker sequence $i\alpha \bmod 1, i=0,1,2,\ldots,$ is quasi-uniform if and only if $\alpha$ is a badly approximable number. Our elementary proof relies on a result on the three-gap…

We give an extension of a criterion of van der Corput on uniform distribution of sequences. Namely, we prove that a sequence $x_n$ is uniformly distributed modulo 1 if it is weakly monotonic and satisfies the conditions $\Delta^2x_n\to…

A well known result in the theory of uniform distribution modulo one (which goes back to Fej\'er and Csillag) states that the fractional parts $\{n^\alpha\}$ of the sequence $(n^\alpha)_{n\ge1}$ are uniformly distributed in the unit…

Given a fixed quadratic extension K of Q, we consider the distribution of elements in K of norm 1 (denoted N). When K is an imaginary quadratic extension, N is naturally embedded in the unit circle in C and we show that it is…

In this paper we prove an effective equidistribution result for both primitive and non-primitive points on certain expanding horocycle sections in ASL$(2,\mathbb{Z}) \backslash $ASL$(2,\mathbb{R})$. This provides an effective version of a…

Let $G=\SL(2,\R)\ltimes(\R^2)^{k}$, let $\Gamma$ be a congruence subgroup of $\SL(2,\Z)\ltimes(\Z^2)^{k}$, and let $u_{\R}=(u_x)_{x\in\R}$ be the one-parameter subgroup of $G$ given by $u_x=\left(\matr 1x01,0\right)$. We prove polynomially…

We study the limiting distribution of the rational points under a horizontal translation along a sequence of expanding closed horocycles on the modular surface. Using spectral methods we confirm equidistribution of these sample points for…

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira on the asymptotic distribution of primitive rational points on expanding closed horospheres in the space of lattices. Key ingredients of our proof include…

We introduce a natural way of associating oriented closed geodesics on the modular curve to elements of $(\mathbb{Z}/q\mathbb{Z})^\times$ and prove that the corresponding packets associated to sufficiently large subgroups equidistribute in…

We prove a quantum ergodicity theorem in position space for the eigenfunctions of a Schr\"odinger operator $-\Delta+V$ on a rectangular torus $\mathbb{T}^2$ for $V\in L^2(\mathbb{T}^2)$ with an algebraic rate of convergence in terms of the…

Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Str\"ombergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G,…

We prove general equidistribution statements (both conditional and unconditional) relating to the Fourier coefficients of arithmetically normalized holomorphic Hecke cusp forms $f_1,\ldots,f_k$ without complex multiplication, of equal…

We prove effective equidistribution of expanding horocycles in $\mathrm{SL}_2(\mathbb{Z})\backslash\mathrm{SL}_2(\mathbb{R})$ with respect to various classes of Borel probability measures on $\mathbb{R}$ having certain Fourier asymptotics.…

Uncertainty principles for generating systems $\{e_n\}_{n=1}^{\infty} \subset \ltwo$ are proven and quantify the interplay between $\ell^r(\N)$ coefficient stability properties and time-frequency localization with respect to $|t|^p$ power…

Koksma's equidistribution theorem from 1935 states that for Lebesgue almost every $\alpha>1$, the fractional parts of the geometric progression $(\alpha^{n})_{n\geq1}$ are equidistributed modulo one. In the present paper we sharpen this…

We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions.…

Given any line in the plane, we strengthen the Littlewood conjecture by two logarithms for almost every point on the line, thereby generalising the fibre result of Beresnevich, Haynes, and Velani. To achieve this, we prove an effective…

For a complete hyperbolic surface whose fundamental group is finitely generated and has critical exponent bigger than 1/2, we obtain an effective equidistribution of closed horocycles in its unit tangent bundle. This extends a result of…

We prove effective equidistribution of non-closed horocycles in the unit tangent bundle of infinite-volume geometrically finite hyperbolic surfaces.