##
**Diophantine approximation on planar curves and the distribution of rational points. With an appendix: Sums of two squares near perfect squares by R. C. Vaughan.**
*(English)*
Zbl 1137.11048

Let \(\psi: \mathbb N \rightarrow \mathbb R_+\) be a non-increasing function. If \(\sum \psi(h)^2 = \infty\), then almost all points \({\mathbf x} \in \mathbb R^2\) satisfy

\[ \| q{\mathbf x}\| < \psi(q), \tag \(*\) \]

for infinitely many \(q \in \mathbb N\), where \(\| {\mathbf x} \|\) denotes the distance to the nearest point with integer coordinates. If on the other hand \(\sum \psi(h)^2 < \infty\), for almost every \({\mathbf x}\), (\(\ast\)) has at most finitely many solutions. This result goes back to M. Khintchine [Math. Z. 24, 706–714 (1926; JFM 52.0183.02)].

In this important paper, the authors study the simultaneous approximation by rational points with the same denominator of points on a \(C^3\) planar curve \({\mathcal C}\), which satisfies certain non-degeneracy conditions. A curve \({\mathcal C}\) is said to be of Khintchine type for divergence (resp. convergence) if the divergence (resp. convergence) of the series \(\sum \psi(h)\) ensures that for almost all (resp. almost no) \(x \in {\mathcal C}\), (\(\ast\)) has infinitely many solutions. Here, the measure is the natural one-dimensional measure on \({\mathcal C}\).

It is shown that if the curvature of \({\mathcal C}\) is non-zero almost everywhere, then \({\mathcal C}\) is of Khintchine type for divergence. Additionally, the more refined question of determining the Hausdorff dimension of the exceptional set in case of convergence is studied when this is in the range \((1/2,1)\). A lower bound is found when the curvature is non-zero at some point. It is shown that the lower bound coincides with an upper bound when the set where the curvature vanishes is of small Hausdorff dimension.

In the case when \({\mathcal C}\) is a rational quadric, i.e., a rational affine transformation of either the standard hyperbola \(x_1^2 - x_2^2 = 1\), the standard parabola \(x_2 = x_1^2\) or the unit circle \(x_1^2 + x_2^2 = 1\), it is shown that \({\mathcal C}\) is also of Khintchine type for convergence. In this case, the Hausdorff dimension of the exceptional set is completely determined when it falls in \((1/2, 1)\).

The divergence results and lower bounds for Hausdorff dimension are derived using the notion of a ubiquitous system. The general framework is that of V. Beresnevich, D. Dickinson and S. Velani [Mem. Am. Math. Soc. 846, 91 p. (2006; Zbl 1129.11031)]. The required results from that paper are proved in a less general form, which is still sufficient for the problem at hand, in Appendix I. For rational quadrics, where the convergence case is also shown, the proof requires an asymptotic estimate for the number of sums of squares near a perfect squares. This estimate is derived in Appendix II, which is written by R. C. Vaughan.

\[ \| q{\mathbf x}\| < \psi(q), \tag \(*\) \]

for infinitely many \(q \in \mathbb N\), where \(\| {\mathbf x} \|\) denotes the distance to the nearest point with integer coordinates. If on the other hand \(\sum \psi(h)^2 < \infty\), for almost every \({\mathbf x}\), (\(\ast\)) has at most finitely many solutions. This result goes back to M. Khintchine [Math. Z. 24, 706–714 (1926; JFM 52.0183.02)].

In this important paper, the authors study the simultaneous approximation by rational points with the same denominator of points on a \(C^3\) planar curve \({\mathcal C}\), which satisfies certain non-degeneracy conditions. A curve \({\mathcal C}\) is said to be of Khintchine type for divergence (resp. convergence) if the divergence (resp. convergence) of the series \(\sum \psi(h)\) ensures that for almost all (resp. almost no) \(x \in {\mathcal C}\), (\(\ast\)) has infinitely many solutions. Here, the measure is the natural one-dimensional measure on \({\mathcal C}\).

It is shown that if the curvature of \({\mathcal C}\) is non-zero almost everywhere, then \({\mathcal C}\) is of Khintchine type for divergence. Additionally, the more refined question of determining the Hausdorff dimension of the exceptional set in case of convergence is studied when this is in the range \((1/2,1)\). A lower bound is found when the curvature is non-zero at some point. It is shown that the lower bound coincides with an upper bound when the set where the curvature vanishes is of small Hausdorff dimension.

In the case when \({\mathcal C}\) is a rational quadric, i.e., a rational affine transformation of either the standard hyperbola \(x_1^2 - x_2^2 = 1\), the standard parabola \(x_2 = x_1^2\) or the unit circle \(x_1^2 + x_2^2 = 1\), it is shown that \({\mathcal C}\) is also of Khintchine type for convergence. In this case, the Hausdorff dimension of the exceptional set is completely determined when it falls in \((1/2, 1)\).

The divergence results and lower bounds for Hausdorff dimension are derived using the notion of a ubiquitous system. The general framework is that of V. Beresnevich, D. Dickinson and S. Velani [Mem. Am. Math. Soc. 846, 91 p. (2006; Zbl 1129.11031)]. The required results from that paper are proved in a less general form, which is still sufficient for the problem at hand, in Appendix I. For rational quadrics, where the convergence case is also shown, the proof requires an asymptotic estimate for the number of sums of squares near a perfect squares. This estimate is derived in Appendix II, which is written by R. C. Vaughan.

Reviewer: Simon Kristensen (Aarhus)