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Flows on homogeneous spaces and Diophantine approximation on manifolds. (English) Zbl 0922.11061

This important paper opens up a new avenue in the theory of metric Diophantine approximation. The authors apply ideas drawn from dynamical systems to prove the Baker-Sprindzhuk conjecture (and more) concerning the strong extremality of analytic manifolds satisfying certain non-degeneracy conditions and which implies Sprindzhuk’s conjecture on extremality. Call a point $x\in {ℝ}^{n}$ very well approximable if for some $\epsilon >0$, there are infinitely many $q\in {ℤ}^{n}$ such that inequality

${|q·x+p|\parallel q\parallel }^{n}\le {\parallel q\parallel }^{-n\epsilon },$

where $\parallel q\parallel$ is the height $max\left\{|{q}_{i}|:i=1,\cdots ,n\right\}$, holds for some $p\in ℤ$.

Using ideas due to the second author and to S. G. Dani, they show that the set of very well approximable points $x$ on a smooth ‘non-degenerate’ manifold has induced Lebesgue measure zero. The proof is based on a correspondence between approximation properties of points in ${ℝ}^{n}$ and the behaviour of certain orbits in the homogeneous space of unimodular lattices in ${ℝ}^{n+1}$. The core of the proof is an improvement on a result for non-divergent unipotent flows on lattices. The argument is rather technical but is clearly explained. The paper concludes with some interesting open questions.

The first author has used similar ideas to establish the inhomogeneous analogue of the result of W. M. Schmidt that the Hausdorff dimension of the set of badly approximable linear forms is maximal (or more precisely, the set is ‘thick’) [D. Kleinbock, Badly approximable systems of affine forms, J. Number Theory 79, No. 1, 83–102 (1999; Zbl 0937.11030)].

##### MSC:
 11J83 Metric theory of numbers 37A17 Homogeneous flows 37A45 Relations of ergodic theory with number theory and harmonic analysis 22E40 Discrete subgroups of Lie groups