Flows on \(S\)-arithmetic homogeneous spaces and applications to metric Diophantine approximation.

*(English)*Zbl 1135.11037A measure \(\mu\) on \(\mathbb R^n\) is said to be extremal if \(\mu\)-almost all points are no better approximable by rational points with the same denominator than generic points in \(\mathbb R^n\). In other words, if the \(\mu\)-measure of the set of points \({\mathbf x} \in \mathbb R^n\) for which the inequality

\[ | q{\mathbf x} - {\mathbf p} | < {1 \over{q^{1/n + \varepsilon}}} \] has infinitely many solutions \(q \in \mathbb N\), \({\mathbf p} \in \mathbb Z^n\) is equal to zero. In a seminal paper, D. Kleinbock and G. A. Margulis [Ann. Math. (2) 148, No. 1, 339–360 (1998; Zbl 0922.11061)] showed that the natural measure on a non-degenerate manifold enjoys this property as well as the property of being strongly extremal, where the usual norm is replaced by a multiplicative variant. Numerous measures have since then been shown to be (strongly) extremal, including e.g. the natural measure on certain fractal sets.

The present paper extends the definition of extremality to the non-Archimedean case. The setup considers the rational approximation of points in \(\mathbb Q_S^n = \prod_{v \in S} \mathbb Q_v^n\), where \(S\) is a finite set of valuations, so that \(\mathbb Q_v\) is the \(p\)-adic numbers when \(v = p < \infty\) and \(\mathbb Q_v = \mathbb R\) when \(v=\infty\). The definitions of extremality and strong extremality are suitably extended to this setup, and it is shown that a large class of measures enjoy these properties.

In addition to the number theory, results which are important in their own right on \(S\)-arithmetic dynamics are derived. For example, a result of S. G. Dani [Ergodic Theory Dyn. Syst. 6, 167–182 (1986; Zbl 0601.22003)] on the finiteness of locally finite ergodic unipotent-invariant measures on homogeneous spaces is extended to the \(S\)-arithmetic setup. The latter result is proved only in the case when \(\infty \in S\).

The proofs of the main results are based on an extension of the methods applied by Kleinbock and Margulis [loc. cit.] to the \(S\)-arithmetic setting. The main results are derived from an \(S\)-arithmetic quantitative non-divergence estimate. A number of complications arise from the \(p\)-adic analysis, where standard results such as the Mean Value Theorem are no longer available. Additionally, setting the scene for the non-divergence result takes a lot of effort.

The paper is well written and to a large extent self-contained. Only a modest amount of background knowledge in \(p\)-adic analysis is needed to understand the proof. The paper is concluded with a nice section on open problems.

\[ | q{\mathbf x} - {\mathbf p} | < {1 \over{q^{1/n + \varepsilon}}} \] has infinitely many solutions \(q \in \mathbb N\), \({\mathbf p} \in \mathbb Z^n\) is equal to zero. In a seminal paper, D. Kleinbock and G. A. Margulis [Ann. Math. (2) 148, No. 1, 339–360 (1998; Zbl 0922.11061)] showed that the natural measure on a non-degenerate manifold enjoys this property as well as the property of being strongly extremal, where the usual norm is replaced by a multiplicative variant. Numerous measures have since then been shown to be (strongly) extremal, including e.g. the natural measure on certain fractal sets.

The present paper extends the definition of extremality to the non-Archimedean case. The setup considers the rational approximation of points in \(\mathbb Q_S^n = \prod_{v \in S} \mathbb Q_v^n\), where \(S\) is a finite set of valuations, so that \(\mathbb Q_v\) is the \(p\)-adic numbers when \(v = p < \infty\) and \(\mathbb Q_v = \mathbb R\) when \(v=\infty\). The definitions of extremality and strong extremality are suitably extended to this setup, and it is shown that a large class of measures enjoy these properties.

In addition to the number theory, results which are important in their own right on \(S\)-arithmetic dynamics are derived. For example, a result of S. G. Dani [Ergodic Theory Dyn. Syst. 6, 167–182 (1986; Zbl 0601.22003)] on the finiteness of locally finite ergodic unipotent-invariant measures on homogeneous spaces is extended to the \(S\)-arithmetic setup. The latter result is proved only in the case when \(\infty \in S\).

The proofs of the main results are based on an extension of the methods applied by Kleinbock and Margulis [loc. cit.] to the \(S\)-arithmetic setting. The main results are derived from an \(S\)-arithmetic quantitative non-divergence estimate. A number of complications arise from the \(p\)-adic analysis, where standard results such as the Mean Value Theorem are no longer available. Additionally, setting the scene for the non-divergence result takes a lot of effort.

The paper is well written and to a large extent self-contained. Only a modest amount of background knowledge in \(p\)-adic analysis is needed to understand the proof. The paper is concluded with a nice section on open problems.

Reviewer: Simon Kristensen (Aarhus)

##### MSC:

11J83 | Metric theory |

11J54 | Small fractional parts of polynomials and generalizations |

11J61 | Approximation in non-Archimedean valuations |

37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |

37D40 | Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) |