arXiv subjects
Publications and source records attributed to Dmitry Kleinbock.
We present a sharpening of nondivergence estimates for unipotent (or more generally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approximation, it yields precise formulas for Diophantine exponents of affine subspaces of $\Bbb R^n$ and their nondegenerate submanifolds.
We show that for any $ε<1$ and any $\mathcal{T}$ `drifting away from walls', Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$ for Lebesgue almost every system of linear forms $Y$ (see the paper for definitions). In the case $m = 1$ we also show that for a large class of measures $μ$ there is $ε_0>0$ such that for any drifting away from walls $\mathcal{T}$, any $ε<ε_0$, and for $μ$-almost every $Y$, Dirichlet's Theorem cannot be $ε$-improved along $\mathcal{T}$. These measures include natural measures on sufficiently regular smooth manifolds and fractals. Our results extend those of several authors beginning with the work of Davenport and Schmidt done in late 1960s. The proofs rely on a translation of the problem into a dynamical one regarding the action of a diagonal semigroup on the space $\text{SL}_{m+n}(\mathbb{R})/\text{SL}_{m+n}(\mathbb{Z})$.
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.
We place the theory of metric Diophantine approximation on manifolds into a broader context of studying Diophantine properties of points generic with respect to certain measures on $\Bbb R^n$. The correspondence between multidimensional Diophantine approximation and dynamics of lattices in Euclidean spaces is discussed in an elementary way, and several recent results obtained by means of this correspondence are surveyed.
We prove that singular vectors have measure zero with respect to any friendly measure on $\Bbb R^n$ (e.g. the volume measure on a nondegenerate submanifold). This generalizes special cases considered by Davenport-Schmidt, Baker and Bugeaud. The main tool is quantitative nondivergence estimates for quasi-polynomial flows on homogeneous spaces.
The paper investigates various $p$-adic versions of Littlewood's conjecture, generalizing a set-up considered recently by de Mathan and Teulie. In many cases it is shown that the sets of exceptions to these conjectures have Hausdorff dimension zero. The proof follows the measure ridigity approach of Einsiedler, Katok and Lindenstrauss.
It is known that the properties of almost all points of R^n being not very well (multiplicatively) approximable are inherited by nondegenerate in R^n (read: not contained in a proper affine subspace) smooth submanifolds. In this paper we consider submanifolds which are contained in proper affine subspaces, and prove that the aforementioned diophantine properties pass from a subspace to its nondegenerate submanifold. The proofs are based on a correspondence between multidimensional diophantine approximation and dynamics of lattices in Euclidean spaces.
In the moduli space of quadratic differentials over complex structures on a surface, we construct a set of full Hausdorff dimension of points with bounded Teichmüller geodesic trajectories.The main tool is quantitative nondivergence of Teichmüller horocycles, due to Minsky and Weiss. This has an application to billiards in rational polygons.
We show a large class of analytic submanifolds of C^n to be strongly extremal. This generalizes V. Sprindzhuk's solution of the complex case of Mahler's Problem, and settles complex analogues of conjectures made in the 1970s by Baker and Sprindzhuk. The proof is based on a variation of quantitative nondivergence estimates for quasi-polynomial flows on the space of lattices.
This survey paper is not a complete reference guide to number-theoretical applications of ergodic theory. Instead, it considers an approach to a class of problems involving Diophantine properties of $n$-tuples of real numbers, namely, describes a specific dynamical system which is naturally connected with these problems.
Let $T: X\mapsto X$ be a deterministic dynamical system preserving a probability measure $μ$. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsets $A_n\subset X$ and $μ$-almost every point $x\in X$ the inclusion $T^nx\in A_n$ holds for infinitely many $n$. We discuss here systems which are either symbolic (topological) Markov chain or Anosov diffeomorphisms preserving Gibbs measures. We find sufficient conditions on sequences of cylinders and rectangles, respectively, that ensure the dynamical Borel-Cantelli lemma.
We prove an inhomogeneous analogue of W. M. Schmidt's (1969) theorem on Hausdorff dimension of the set of badly approximable systems of linear forms. The proof is based on ideas and methods from the theory of dynamical systems, in particular, on abundance of bounded orbits of mixing flows on homogeneous spaces of Lie groups.
We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces. This approach yields a new proof of a conjecture of Mahler, originally settled by V. Sprindzhuk in 1964. We also prove several related hypotheses of A. Baker and V. Sprindzhuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.