Dmitry Kleinbock

Flows on homogeneous spaces and Diophantine approximation on manifolds

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. G. Sprindzuk in 1964. We also prove several related hypotheses of Baker and Sprindzuk formulated in 1970s. The core of the proof is a theorem which generalizes and sharpens earlier results on nondivergence of unipotent flows on the space of lattices.

Logarithm laws for flows on homogeneous spaces

Abstract.In this paper we generalize and sharpen D. Sullivan’s logarithm law for geodesics by specifying conditions on a sequence of subsets {At | t∈ℕ} of a homogeneous space G/Γ (G a semisimple Lie group, Γ an irreducible lattice) and a sequence of elements ft of G under which #{t∈ℕ | ftx∈At} is infinite for a.e. x∈G/Γ. The main tool is exponential decay of correlation coefficients of smooth functions on G/Γ. Besides the general (higher rank) version of Sullivan’s result, as a consequence we obtain a new proof of the classical Khinchin-Groshev theorem on simultaneous Diophantine approximation, and settle a conjecture recently made by M. Skriganov.

Khintchine-type theorems on manifolds: the convergence case for standard and multiplicative versions

Notation. The main objects of this paper are n-tuples y = (y1, . . . , yn) of real numbers viewed as linear forms, i.e. as row vectors. In what follows, y will always mean a row vector, and we will be interested in values of a linear form given by y at integer points q = (q1, . . . , qn)T , the latter being a column vector. Thus yq will stand for y1q1 + · · ·+ ynqn. Hopefully it will cause no confusion. We will study differentiable maps f = (f1, . . . , fn) from open subsets U of R to R; again, f will be interpreted as a row vector, so that f(x)q stands for q1f1(x) + · · · + qnfn(x). In contrast, the elements of the “parameter set” U will be denoted by x = (x1, . . . , xd) without boldfacing, since the linear structure of the parameter space is not significant. For f as above we will denote by ∂if : U 7→ R, i = 1, . . . , d, its partial derivative (also a row vector) with respect to xi. If F is a scalar function on U , we will denote by∇F the column vector consisting of partial derivatives of F . With some abuse of notation, the same way we will treat vector functions f : namely, ∇f will stand for the matrix function U 7→ Md×n(R) with rows given by partial derivatives ∂if . We will also need higher order differentiation: for a multiindex β = (i1, . . . , id), ij ∈ Z+, we let |β| = ∑d j=1 ij and ∂β = ∂ i1 1 ◦ · · · ◦ ∂ id d . Unless otherwise indicated, the norm ‖x‖ of a vector x ∈ R (either row or column vector) will stand for ‖x‖ = max1≤i≤k |xi|. In some cases however we will work with the Euclidean norm ‖x‖ = ‖x‖e = √∑k i=1 x 2 i , keeping the same notation. This distinction will be clearly emphasized to avoid confusion. We will denote by R1 the set of unit vectors in R (with respect to the Euclidean norm). We will use the notation |〈x〉| for the distance between x ∈ R and the closest integer, |〈x〉| def = mink∈Z |x− k|. (It is quite customary to use ‖x‖ instead, but we are not going to do this in order to save the latter notation for norms in vector spaces.) If B ⊂ R, we let |B| stand for the Lebesgue measure of B.

Dynamical Borel-Cantelli lemmas for gibbs measures

LetT: X→X be a deterministic dynamical system preserving a probability measure μ. A dynamical Borel-Cantelli lemma asserts that for certain sequences of subsetsAn⊃ X and μ-almost every pointx∈X the inclusionTnx∈An holds for infinitely manyn. 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.

Flows on S-arithmetic homogeneous spaces and applications to metric Diophantine approximation

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 Rn) are generalized to the S-arithmetic setting.

Badly approximable vectors on fractals

For a large class of closed subsetsC of ℝn, we show that the intersection ofC with the set of badly approximable vectors has the same Hausdorff dimension asC. The sets are described in terms of measures they support. Examples include (but are not limited to) self-similar sets such as Cantor’s ternary sets or attractors for irreducible systems of similarities satisfying Hutchinson’s open set condition.

An extension of quantitative nondivergence and applications to Diophantine exponents

We present a sharpening of nondivergence estimates for unipotent (or more gener- ally polynomial-like) flows on homogeneous spaces. Applied to metric Diophantine approxima- tion, it yields precise formulas for Diophantine exponents of affine subspaces ofR n and their nondegenerate submanifolds.

THE SET OF BADLY APPROXIMABLE VECTORS IS STRONGLY C 1 INCOMPRESSIBLE

We prove that the countable intersection of C 1 -diffeomorphic images of cer- tain Diophantine sets has full Hausdorff dimension. For example, we show this for the set of badly approximable vectors in R d , improving earlier results of Schmidt and Dani. To prove this, inspired by ideas of McMullen, we define a new variant of Schmidts (�,�)-game and show that our sets are hyperplane absolute winning (HAW), which in particular implies winning in the original game. The HAW property passes automati- cally to games played on certain fractals, thus our sets intersect a large class of fractals in a set of positive dimension. This extends earlier results of Fishman to a more general set-up, with simpler proofs. ∞ \ i=1 f −1 i (S)

FLOWS ON HOMOGENEOUS SPACES AND DIOPHANTINE PROPERTIES OF MATRICES

We generalize the notions of badly approximable (resp. singular) systems of m linear forms in n variables, and relate these generalizations to certain bounded (resp. divergent) trajectories in the space of lattices in Rm+n.

Schmidt’s game, fractals, and orbits of toral endomorphisms

Given an integer matrix M ∈GL n (ℝ) and a point y ∈ℝ n /ℤ n , consider the set S. G. Dani showed in 1988 that whenever M is semisimple and y ∈ℚ n /ℤ n , the set has full Hausdorff dimension. In this paper we strengthen this result, extending it to arbitrary M ∈GL n (ℝ)∩M n × n (ℤ) and y ∈ℝ n /ℤ n , and in fact replacing the sequence of powers of M by any lacunary sequence of (not necessarily integer) m × n matrices. Furthermore, we show that sets of the form and their generalizations always intersect with ‘sufficiently regular’ fractal subsets of ℝ n . As an application, we give an alternative proof of a recent result [M. Einsiedler and J. Tseng. Badly approximable systems of affine forms, fractals, and Schmidt games. Preprint , arXiv:0912.2445] on badly approximable systems of affine forms.