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Rational points near manifolds and metric Diophantine approximation. (English) Zbl 1264.11063

In this important paper, spectacular progress in the metric theory of Diophantine approximation on manifolds is made. The author proves that an analytic and non-degenerate sub-manifold \({\mathcal M} \subseteq {\mathbb R}^n\) is of Khintchine type for divergence. This means that if \(\psi : {\mathbb N} \rightarrow {\mathbb R}_+\) is a decreasing function with \(\sum \psi(k)^n = \infty\), then for almost every point \({\mathbf x} \in {\mathcal M}\) with respect to the natural measure on \({\mathcal M}\), the inequality \[ |q{\mathbf x}-{\mathbf p}| < \psi(q) \] has infinitely many solutions \(q \in {\mathbb Z}, {\mathbf p} \in {\mathbb Z}^n\). Previously, this result was known only for \(C^{(3)}\) planar curves. At the cost of imposing analyticity, this is extended to all non-degenerate sub-manifolds of \({\mathbb R}^n\).
The key difficulty in establishing results as the above lie in calculating the contribution from rational points with the same denominator nearby the manifold in question. In particular, one needs to obtain a good lower bound for the number of rational points within a specified distance of \({\mathcal M}\), with bounded denominator \(q\) and which all lie in some specified set \(B\). The derivation of such a bound constitutes the bulk of the paper, which consists of a tour de force of calculations involving geometric and probabilistic ideas. Despite the technical nature of the problems involved, the paper remains very readable and self-contained.
In addition to the statement on manifolds being of Khintchine type for divergence, the author also derives the Hausdorff measure of some exceptional sets as well as even stronger results in the case of curves.
The paper is concluded with some conjectures. One states that analytic and non-degenerate manifolds should be of Khintchine type for convergence, i.e., if the series \(\sum \psi(k)^n\) converges, the set of points for which the above inequality is satisfied infinitely often is of measure \(0\). Another conjecture gives the corresponding Hausdorff measure statement. Both of these conjectures would involve obtaining an upper bound on the number of rational points nearby the manifold, which would complement the lower bound obtained in this paper.

MSC:

11J83 Metric theory
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