Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds

Abstract

Abstract. 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. Sprindžuk in 1964. We also prove several related hypotheses of Baker and Sprindžuk 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.

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

“…[10,13,27]. We actually make use of [27] to control how much mass of a closed H-orbit can be close to infinity, see Lemma 3.6.1.…”

“…We shall refer to ι 4 ht(x) −κ 3 as the injectivity radius at x. We shall require the following lemma, which relies on the mentioned linearization technique [27] and on our technical assumption in §1.2 that the centralizer of h is trivial (in the form of Lemma 3.4.1). The proof is given in Appendix B.…”

“…Consequently, for T sufficiently large, the s i are 1 20 -separated and belong to B S (c 2 T ). 27 We are going to apply the upper bound (part 2) of Proposition 12.2 with Λ as in (13.7), B := B S (c 2 T ) and with {s i } as the separated subset of B ∩ Λs 1 . By the remark after Proposition 12.2, the proposition also yields bounds for the cardinality of any 1 20 -separated subset, such as {s i }; the upper bound is weaker by a constant that depends only on H, G. Notations as in that proposition, the ball "B" is contained in B S (c 3 T ), for T sufficiently large, as we see by applying property (3) of §3.5.…”

“…What we will use and actually need for (2) of Lemma 3.6.1 is a quantification of this phenomenon. We use results from the paper [27] by Kleinbock and G.M.…”

Exceptional Sets in Dynamical Systems and Diophantine Approximation

“…Such points are called very well multiplicatively approximable or VWMA. The 'strong extremality' conjecture was proved by Kleinbock and Margulis [50] for smooth manifolds which are non-degenerate (or 'non-flat' locally) almost everywhere. Their proof uses actions on the lattice…”