Related papers: An Effective Ratner Equidistribution Result for AS…

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact…

Let $\Gamma$ be a lattice of a semisimple Lie group $L$. Suppose that one parameter Ad-diagonalizable subgroup $\{g_t\}$ of $L$ acts ergodically on $L/\Gamma$ with respect to the probability Haar measure $\mu$. For certain proper subgroup…

Let G be a semisimple Lie group without compact factor and $\Gamma$ < G a torsion-free, cocompact, irreducible lattice. According to Selberg, periodic orbits of regular Weyl chamber flows live on tori. We prove that these periodic tori…

We prove a number field analogue of W. M. Schmidt's conjecture on the intersection of weighted badly approximable vectors and use this to prove an instance of a conjecture of An, Guan and Kleinbock. Namely, let $G := SL_2(\mathbb{R}) \times…

We show that there is a consistent polynomial quantization of the coordinate ring of a basic nilpotent coadjoint orbit of a semisimple Lie group. We also show, at least in the case of a nilpotent orbit in sl(2,R)*, that any such…

We study the limiting distributions of expanding translates of a compact segment of a smooth curve under a diagonal subgroup of $G=\mathrm{SO}(n_1,1)\times\cdots\times\mathrm{SO}(n_k,1)$, where $G$ acts on a finite volume homogeneous space…

We describe a short, reproducible workflow for applying finite differences on nonuniform grids determined by a positive weight function g. The grid is obtained by equidistribution, mapping uniform computational coordinates $\xi\in[0,1]$ to…

Given a sequence of complex numbers {\rho}_n, we study the asymptotic distribution of the sets of parameters c {\epsilon} C such that the quadratic maps z^2 +c has a cycle of period n and multiplier {\rho}_n. Assume 1/n.log|{\rho}_n| tends…

Let $G_n(z)=\xi_0+\xi_1z+...+\xi_n z^n$ be a random polynomial with i.i.d. coefficients (real or complex). We show that the arguments of the roots of $G_n(z)$ are uniformly distributed in $[0,2\pi]$ asymptotically as $n\to\infty$. We also…

Let \Gamma<\PSL(2,\C) be a geometrically finite non-elementary discrete subgroup, and let its critical exponent \delta\ be greater than 1. We use representation theory of \PSL(2,\C) to prove an effective bisector counting theorem for…

For a finite group G, we study the higher commuting probabilities, namely the probabilities that r randomly chosen elements of G commute pairwise, together with the corresponding numbers of simultaneous conjugacy classes of commuting…

The present paper is concerned with equidistribution results for certain flows on homogeneous spaces and related questions in Diophantine approximation. Firstly, we answer in the affirmative, a question raised by Kleinbock, Shi and Weiss…

Let $M$ be a closed Riemannian manifold carrying an effective and isometric action of a compact connected Lie group $G$. We derive a refined remainder estimate in the stationary phase approximation of certain oscillatory integrals on…

We study matrix coefficients of the unitary (and also the completely bounded) representations of SL(2;R) and its universal covering group. We describe the asymptotic distribution of column vectors in terms of Whittaker functions, exhibiting…

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…

Let $M=\Gamma\backslash\mathrm{PSL}(2,\mathbb{R})$ be a compact manifold, and let $f\in C^\infty(M)$ be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of $f$…

We prove a new equidistribution estimate for the divisor function in arithmetic progression to moduli that have two small factors. We give two applications. First, we show an asymptotic formula for the divisor function over arithmetic…

We answer a question of Katona and Makar-Limanov, by showing that in an abelian group of order $2h$ the $h$-element subset sums are asymptotically (as $h\to \infty$) equidistributed. In fact we prove a more general result where the order of…

Let $\mathbb F_q$ be the finite field of $q$ elements having characteristic $p$, and denote by $\mathbb K_\infty=\mathbb F_q((1/t))$ the field of formal Laurent series in $1/t$. We consider the equidistribution in $\mathbb T=\mathbb…

We derive an asymptotic equation for quasi-static, nonlinear surface plasmons propagating on a planar interface between isotropic media. The plasmons are nondispersive with a constant linearized frequency that is independent of their…