Expanding translates of curves and Dirichlet-Minkowski theorem on linear forms

We show that a multiplicative form of Dirichlet's theorem on simultaneous Diophantine approximation as formulated by Minkowski, cannot be improved for almost all points on any analytic curve on R^k which is not contained in a proper affine subspace. Such an investigation was initiated by Davenport and Schmidt in the late sixties. The Diophantine problem is then settled by showing that certain sequence of expanding translates of curves on the homogeneous space of unimodular lattices in R^{k+1} gets equidistributed in the limit. We use Ratner's theorem on unipotent flows, linearization techniques, and a new observation about intertwined linear dynamics of various SL(m,R)'s contained in SL(k+1,R).Comment: 28 page

Counting integral matrices with a given characteristic polynomial

We give a simpler proof of an earlier result giving an asymptotic estimate for the number of integral matrices, in large balls, with a given monic integral irreducible polynomial as their common characteristic polynomial. The proof uses equidistributions of polynomial trajectories on SL(n,R)/SL(n,Z), which is a generalization of Ratner's theorem on equidistributions of unipotent trajectories. We also compute the exact constants appearing in the above mentioned asymptotic estimate

Limiting distributions of curves under geodesic flow on hyperbolic manifolds

We consider the evolution of a compact segment of an analytic curve on the unit tangent bundle of a finite volume hyperbolic $n$-manifold under the geodesic flow. Suppose that the curve is not contained in a stable leaf of the flow. It is shown that under the geodesic flow, the normalized parameter measure on the curve gets asymptotically equidistributed with respect to the normalized natural Riemannian measure on the unit tangent bundle of a closed totally geodesically immersed submanifold. Moreover, if this immersed submanifold is a proper subset, then a lift of the curve to the universal covering space $T^1(H^n)$ is mapped into a proper subsphere of the ideal boundary sphere $\partial H^n$ under the visual map. This proper subsphere can be realized as the ideal boundary of an isometrically embedded hyperbolic subspace in $H^n$ covering the closed immersed submanifold. In particular, if the visual map does not send a lift of the curve into a proper subsphere of $\partial H^n$, then under the geodesic flow the curve gets asymptotically equidistributed on the unit tangent bundle of the manifold with respect to the normalized natural Riemannian measure. The proof uses dynamical properties of unipotent flows on finite volume homogeneous spaces of SO(n,1).Comment: 27 pages, revised version, Proof of Theorem~3.1 simplified, remarks adde

Geometric results on linear actions of reductive Lie groups for applications to homogeneous dynamics

Several problems in number theory when reformulated in terms of homogenous dynamics involve study of limiting distributions of translates of algebraically defined measures on orbits of reductive groups. The general non-divergence and linearization techniques, in view of Ratner's measure classification for unipotent flows, reduce such problems to dynamical questions about linear actions of reductive groups on finite dimensional vectors spaces. This article provides general results which resolve these linear dynamical questions in terms of natural group theoretic or geometric conditions

Expanding translates of shrinking submanifolds in homogeneous spaces and Diophantine approximation

On the space $\mathcal{L}_{n+1}$ of unimodular lattices in $\mathbb{R}^{n+1}$, we consider the action of $a(t)={\rm diag}(t^n,t^{-1},\ldots,t^{-1})\in {\rm SL}(n+1,\mathbb{R})$ for $t>1$. Let $M$ be a nondegenerate $C^{n+1}$-submanifold of an expanding horospherical leaf in $\mathcal{L}_{n+1}$. We prove that for almost every $x\in M$, the shrinking balls in $M$ of radii $t^{-1}$ around $x$ get asymptotically equidistributed in $\mathcal{L}_{n+1}$ under the action of $a(t)$ as $t\to\infty$. This result implies non-improvability of Dirichlet's Diophantine approximation theorem for almost every point on a nondegenerate $C^{n+1}$-submanifold of $\mathbb{R}^n$, answering a question of Davenport and Schmidt (1969).Comment: 16 page

An upper bound of the Hausdorff dimension of singular vectors on affine subspaces

In Diophantine approximation, the notion of singular vectors was introduced by Khintchine in the 1920's. We study the set of singular vectors on an affine subspace of $\mathbb{R}^n$. We give an upper bound of its Hausdorff dimension in terms of the Diophantine exponent of the parameter of the affine subspace.Comment: 18 page

Equidistribution in the space of 3-lattices and Dirichlet-improvable vectors on planar lines

Let $X=\mathrm{SL}_3(\mathbb{R})/\mathrm{SL}_3(\mathbb{Z})$, and $g_t=\mathrm{diag}(e^{2t}, e^{-t}, e^{-t})$. Let $\nu$ denote the push-forward of the normalized Lebesgue measure on a straight line segment in the expanding horosphere of $\{g_t\}_{t>0}$ by the map $h\mapsto h\mathrm{SL}_3(\mathbb{Z})$, $\mathrm{SL}_3(\mathbb{R})\to X$. We explicitly give necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families in $X$: (1) $g_t$-translates of $\nu$ as $t\to\infty$. (2) averages of $g_t$-translates of $\nu$ over $t\in[0,T]$ as $T\to\infty$. (3) $g_{t_i}$-translates of $\nu$ for some $t_i\to\infty$. We apply the dynamical result to show that Lebesgue-almost every point on the planar line $y=ax+b$ is not Dirichlet-improvable if and only if $(a,b)\notin\mathbb{Q}^2$.Comment: 30 pages, 2 figure

Strong wavefront lemma and counting lattice points in sectors

We compute the asymptotics of the number of integral quadratic forms with prescribed orthogonal decompositions and, more generally, the asymptotics of the number of lattice points lying in sectors of affine symmetric spaces. A new key ingredient in this article is the strong wavefront lemma, which shows that the generalized Cartan decomposition associated to a symmetric space is uniformly Lipschitz