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Nimish A. Shah

Publications and source records attributed to Nimish A. Shah.

18 recordsLinked to original sources
arXiv2026

Birkhoff genericity on affine subspaces in horospheres

We study Birkhoff genericity for a simple uniformly expanding diagonal flow on $\mathrm{SL}_{n+1}(\mathbb R)/\mathrm{SL}_{n+1}(\mathbb Z)$, with initial points restricted to affine subspaces of the expanding horospherical orbit through the identity coset. We prove that almost every point on such an affine subspace is Birkhoff generic, except possibly in two situations: either the defining matrix of the affine subspace has Diophantine exponent at least $n$, or the affine subspace is arbitrarily well approximable by affine subspaces of dimension $(r-1)$ defined over a real number field of degree $m\ge 2$, with $n+1=mr$. As applications, we obtain Dirichlet non-improvability and logarithmic density results for almost every point on these affine subspaces.

math.DS↗
arXiv2026

Limit distributions for $\text{SO}(n,1)$ action on $k$-lattices in $\mathbb{R}^{n+1}$

We study the asymptotic distribution of norm ball averages along orbits of a lattice $Γ\subset \text{SO}(n,1)$ acting on the moduli space of pairs of orthogonal discrete subgroups of $\mathbb{R}^{n+1}$ up to homothety. Our main result shows that, except for special $2$-lattices in $\mathbb{R}^3$ lying in hyperplanes tangent to the light cone, these measures converge to an explicit semi-invariant probability measure supported on the space of homothety classes of pairs of orthogonal lattices tangent to the light cone. Our main motivation is a conjecture of Sargent and Shapira, which is resolved as a special case of our general result.

math.DS↗
arXiv2025

Equidistribution of polynomially bounded o-minimal curves in homogeneous spaces

We extend Ratner's theorem on equidistribution of individual orbits of unipotent flows on finite volume homogeneous spaces of Lie groups to trajectories of non-contracting curves definable in polynomially bounded o-minimal structures. To be precise, let $φ:[0,\infty)\to \text{SL}(n,\mathbb R)$ be a continuous map whose coordinate functions are definable in a polynomially bounded o-minimal structure; for example, rational functions. Suppose that $φ$ is non-contracting; that is, for any linearly independent vectors $v_1,\ldots,v_k$ in $\mathbb R^n$, $φ(t).(v_1\wedge\cdots\wedge v_k)\not\to0$ as $t\to\infty$. Then, there exists a unique smallest subgroup $H_φ$ of $\text{SL}(n,\mathbb R)$ generated by unipotent one-parameter subgroups such that $φ(t)H_φ\to g_0H_φ$ in $\text{SL}(n,\mathbb R)/H_φ$ as $t\to\infty$ for some $g_0\in \text{SL}(n,\mathbb R)$. Let $G$ be a closed subgroup of $\text{SL}(n,\mathbb R)$ and $Γ$ be a lattice in $G$. Suppose that $φ([0,\infty))\subset G$. Then $H_φ\subset G$, and for any $x\in G/Γ$, the trajectory $\{φ(t)x:t\in [0,T]\}$ gets equidistributed with respect to the measure $g_0μ_{Lx}$ as $T\to\infty$, where $L$ is a closed subgroup of $G$ such that $\overline{Hx}=Lx$ and $Lx$ admits a unique $L$-invariant probability measure, denoted by $μ_{Lx}$. A crucial new ingredient in this work is proving that for any finite-dimensional representation $V$ of $\text{SL}(n,\mathbb R)$, there exist $T_0>0$, $C>0$, and $α>0$ such that for any $v\in G$, the map $t\mapsto \|φ(t)v\|$ is $(C,α)$-good on $[T_0,\infty)$.

math.DS↗
arXiv2024

Equidistribution of expanding degenerate manifolds in the space of lattices

For the space of unimodular lattices in a Euclidean space, we give necessary and sufficient conditions for equidistribution of expanding translates of any real-analytic submanifold under a diagonal flow. This extends the earlier result of Shah in the case of non-degenerate submanifolds. We apply the above dynamical result to show that if the affine span of a real-analytic submanifold in a Euclidean space satisfies certain Diophantine and arithmetic conditions, then almost every point on the manifold is not Dirichlet-improvable.

math.DS↗
arXiv2023

Limit distributions of expanding translates of shrinking submanifolds and non-improvability of Dirichlet's approximation theorem

On the space $\mathcal{L}_{n+1}$ of unimodular lattices in $\mathbb{R}^{n+1}$, we consider the standard action of $a(t)=\mathrm{diag}(t^n,t^{-1},\ldots,t^{-1})\in \mathrm{SL}(n+1,\mathbb{R})$ for $t>1$. Let $M$ be a nondegenerate submanifold of an expanding horospherical leaf in $\mathcal{L}_{n+1}$. We prove that for all $x\in M\setminus E$ and $t>1$, if $μ_{x,t}$ denotes the normalized Lebesgue measure on the ball of radius $t^{-1}$ around $x$ in $M$, then the translated measure $a(t)μ_{x,t}$ get equidistributed $\mathcal{L}_{n+1}$ as $t\to\infty$, where $E$ is a union of countably many lower dimensional submanifolds of $M$. In particular, if $μ$ is an absolutely continuous probability measure on $M$, then $a(t)μ$ gets equidistributed in $\mathcal{L}_{n+1}$ as $t\to\infty$. This result implies the non-improvability of Dirichlet's Diophantine approximation theorem for almost every point on a $C^{n+1}$-submanifold of $\mathbb{R}^n$ satisfying a non-degeneracy condition, answering a question arising from the work of Davenport and Schmidt (1969).

math.DS↗
arXiv2022

Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation

We show that under the action of $\mathrm{diag}(e^{nt},e^{-r_1(t)},\ldots,e^{-r_n(t)})\in\mathrm{SL}(n+1,\mathbb{R})$, where $r_i(t)\to\infty$, on the space of unimodular lattices in $\mathbb{R}^{n+1}$, the translates of any fixed-sized piece of a `non-degenerate' smooth curve, or a shrinking piece of size $e^{-t}$ about almost any point of the curve, get equidistributed in the space as $t\to\infty$. From this, it follows that the weighted Dirichlet approximation theorem cannot be improved for almost all points on any non-degenerate $C^{2n}$ curve in $\mathbb{R}^n$. This result extends the corresponding result for analytic curves due to Shah (2009) and answers some questions inspired by the work of Davenport and Schmidt (1969) and Kleinbock and Weiss (2008).

math.DS↗
arXiv2022

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

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

math.DS↗
arXiv2016

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.

math.RT↗
arXiv2008

Asymptotic evolution of smooth curves under geodesic flow on hyperbolic manifolds - II

Extending the earlier results for analytic curve segments, in this article we describe the asymptotic behaviour of evolution of a finite segment of a C^n-smooth curve under the geodesic flow on the unit tangent bundle of a finite volume hyperbolic n-manifold. In particular, we show that if the curve satisfies certain natural geometric conditions, the pushforward of the parameter measure on the curve under the geodesic flow converges to the normalized canonical Riemannian measure on the tangent bundle in the limit. We also study the limits of geodesic evolution of shrinking segments. We use Ratner's classification of ergodic invariant measures for unipotent flows on homogeneous spaces of SO(n,1), and an observation relating local growth properties of smooth curves and dynamics of linear SL(2,R)-actions.

math.DG↗
arXiv2008

Khinchin theorem for integral points on quadratic varieties

We prove an analogue the Khinchin theorem for the Diophantine approximation by integer vectors lying on a quadratic variety. The proof is based on the study of a dynamical system on a homogeneous space of the orthogonal group. We show that in this system, generic trajectories visit a family of shrinking subsets infinitely often.

math.NT↗
arXiv2008

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).

math.NT↗
arXiv2008

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).

math.DG↗
arXiv2008

Equidistribution of expanding translates of curves and Dirichlet's theorem on Diophantine approximation

We show that for almost all points on any analytic curve on R^{k} which is not contained in a proper affine subspace, the Dirichlet's theorem on simultaneous approximation, as well as its dual result for simultaneous approximation of linear forms, cannot be improved. The result is obtained by proving asymptotic equidistribution of evolution of a curve on a strongly unstable leaf under certain partially hyperbolic flow on the space of unimodular lattices in R^{k+1}. The proof involves ergodic properties of unipotent flows on homogeneous spaces.

math.NT↗
arXiv2007

Unipotent flows on products of $SL(2,K)/Γ$'s

We give a simplified and a direct proof of a special case of Ratner's theorem on closures and uniform distribution of individual orbits of unipotent flows; namely, the case of orbits of the diagonally embedded unipotent subgroup acting on $SL(2,K)/Γ_1\times ...\times SL(2,K)/Γ_n$, where $K$ is a locally compact field of characteristic 0 and each $Γ_i$ is a cocompact discrete subgroup of $SL(2,K)$. This special case of Ratner's theorem plays a crucial role in the proofs of uniform distribution of Heegner points by Vatsal, and Mazur conjecture on Heegner points by C. Cornut; and their generalizations in their joint work on CM-points and quaternion algebras. A purpose of the article is to make the ergodic theoretic results accessible to a wide audience.

math.RT↗
arXiv2000

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.

math.RT↗
arXiv2000

Invariant Measures and Orbit Closures on Homogeneous Spaces for Actions of Subgroups Generated by Unipotent Elements

The theorems of M. Ratner, describing the finite ergodic invariant measures and the orbit closures for unipotent flows on homogeneous spaces of Lie groups, are extended for actions of subgroups generated by unipotent elements. More precisely: Let G be a Lie group (not necessarily connected) and Gamma a closed subgroup of G. Let W be a subgroup of G such that Ad(W) is contained in the Zariski closure (in the group of automorphisms of the Lie algebra of G) of the subgroup generated by the unipotent elements of Ad(W). Then any finite ergodic invariant measure for the action of W on G/Gamma is a homogeneous measure (i.e., it is supported on a closed orbit of a subgroup preserving the measure). Moreover, if G/Gamma has finite volume (i.e., has a finite G-invariant measure), then the closure of any orbit of W on G/Gamma is a homogeneous set (i.e., a finite volume closed orbit of a subgroup containing W). Both the above results hold if W is replaced by any subgroup Lambda of W such that W/Lambda has finite volume.

math.RT↗
arXiv2000

Limit distributions of expanding translates of certain orbits on homogeneous spaces

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 subgroup. Let U be the expanding horospherical subgroup of G associated to c. Let Omega be a nonempty open subset of U, and {n_i} be a sequence of natural numbers tending to infinity. It is shown that the image of the union of the sets c^{n_i}.Omega in L/Lambda is dense. A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describe G-equivariant topological factors of the product space L/Lambda \times G/P, where the real rank of G is greater than 1, P is a parabolic subgroup of G, and G acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.

math.RT↗