Related papers: Joint effective equidistribution of partial lattic…

Given a place $\omega$ of a global function field $K$ over a finite field, with associated affine function ring $R_\omega$ and completion $K_\omega$, the aim of this paper is to give an effective joint equidistribution result for…

We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its…

Let $H < G$ both be noncompact connected semisimple real algebraic groups where the former is maximal proper and $\Gamma < G$ be a lattice. Building on the work of Gorodnik-Weiss, we refine their techniques and obtain effective results.…

We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of perfect Lie groups with a polynomial error term. Even for semisimple quotients, our result provides the first infinite…

To any $k$-dimensional subspace of $\mathbb Q^n$ one can naturally associate a point in the Grassmannian ${\rm Gr}_{n,k}(\mathbb R)$ and two shapes of lattices of rank $k$ and $n-k$ respectively. These lattices originate by intersecting the…

Given a local field $\widehat K$ with positive characteristic, we study the dynamics of the diagonal subgroup of the linear group $\operatorname{GL}_n(\widehat K)$ on homogeneous spaces of discrete lattices in ${\widehat K}^{\,n}$. We first…

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…

Let $\Gamma < G$ be an arithmetic lattice in a noncompact connected semisimple real algebraic group. For many such $G$ of rank at most $2$, in particular $G = \operatorname{SL}_3(\mathbb R)$, we prove effective equidistribution of large…

Consider $G=\SL_{ d }(\mathbb R)$ and $ \Gamma=\SL_{ d }(\mathbb Z)$. It was recently shown by the second-named author \cite{s} that for some diagonal subgroups $\{g_t\}\subset G$ and unipotent subgroups $U\subset G$, $g_t$-trajectories of…

Let $G=SL_2(\mathbb R)^d$ and $\Gamma=\Gamma_0^d$ with $\Gamma_0$ a lattice in $SL_2(\mathbb R)$. Let $S$ be any "curved" submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an $\mathbb R$-diagonalizable…

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 study the distribution of orbits of a lattice $\Gamma\leq\text{SL}(3,\mathbb R)$ in the moduli space $X_{2,3}$ of covolume one rank-two discrete subgroups in $\mathbb R^3$. Each orbit is dense, and our main result is the limiting…

We prove the first case of polynomially effective equidistribution of closed orbits of semisimple groups with nontrivial centralizer. The proof relies on uniform spectral gap, builds on, and extends work of Einsiedler, Margulis, and…

We prove a polynomially effective equidistribution result for expanding translates in the space of $d$-dimensional affine lattices for any $d\ge 2$.

We prove the following result: Let K be a lattice, let D be a distributive lattice with zero, and let $\phi$: Con K $\to$ D be a {&#8744;, 0}-homomorphism, where Conc K denotes the {&#8744;, 0}-semilattice of all &#64257;nitely generated…

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 sketch the proof of an effective equidistribution theorem for one-parameter unipotent subgroups in $S$-arithmetic quotients arising from $\mathbf K$-forms of $\mathrm{SL}_2^{\mathsf n}$ where $\mathbf K$ is a number field. This gives an…

Join-distributive lattices are finite, meet-semidistributive, and semimodular lattices. They are the same as Dilworth's lattices in 1940, and many alternative definitions and equivalent concepts have been discovered or rediscovered since…

We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb…

Let $G=$SL$(2,R)\ltimes(R^2)^{\oplus k}$ and let $\Gamma$ be a congruence subgroup of SL$(2,Z)\ltimes(Z^2)^{\oplus k}$. We prove a polynomially effective asymptotic equidistribution result for special types of unipotent orbits in…