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Zbl 1168.11029
Badziahin, Dzmitry; Levesley, Jason
(Bodyagin, D. A.)
A note on simultaneous and multiplicative Diophantine approximation on planar curves. (English)
[J] Glasg. Math. J. 49, No. 2, 367-375 (2007). ISSN 0017-0895; ISSN 1469-509X/e

The authors consider two different forms of simultaneous Diophantine approximation on certain non-degenerate planar curves. Specifically they consider graphs of functions $f : I \rightarrow {\Bbb R}$ where $I \subseteq {\Bbb R}$ is an interval, which are three times continuously differentiable, and which for Lebesgue almost every point in $I$ have derivative bounded away from zero and non-zero second derivative. Denote the graph of such a function by ${\cal C}_f$. \par For a real number $x$, let $\Vert x \Vert$ denote the distance from $x$ to the nearest integer. For decreasing functions $\psi, \psi_1, \psi_2 : {\Bbb R}_+ \rightarrow {\Bbb R}_+$, consider the set ${\cal S}_2^*(\psi)$ of points $(x,y) \in {\Bbb R}^2$ for which the inequality $$ \Vert qx \Vert \Vert qy \Vert < \psi(q) $$ has infinitely many solutions $q \in {\Bbb Z}$ and the set ${\cal S}_2(\psi_1, \psi_2)$ of points $(x,y) \in {\Bbb R}^2$ for which the system of inequalities $$ \Vert q x \Vert < \psi_1(q), \quad \Vert q y \Vert < \psi_2(q) $$ has infinitely many solutions $q \in {\Bbb Z}$. \par It is shown that if $0 < s \leq 1$ and $\sum_{h=1}^\infty h^{1-s} (\log h)^s \psi(h)^s < \infty$, then the Hausdorff $s$-measure of ${\cal S}_2^*(\psi) \cap {\cal C}_f$ is equal to zero. This proves a recent conjecture of {\it R. C. Vaughan} and {\it S. Velani} [Invent. Math. 166, No. 1, 103--124 (2006; Zbl 1185.11047)]. Additionally, it is shown that if $\sum_{h=1}^\infty \psi_1(h) \psi_2(h) < \infty$, then the set ${\cal S}_2(\psi_1, \psi_2) \cap {\cal C}_f$ is null with respect to the natural (Lebesgue) measure on ${\cal C}_f$. \par The proof uses a covering argument together with an asymptotic formula due to Vaughan and Velani [{\it loc.cit.}] for the number of rational points nearby non-degenerate planar curves.
[Simon Kristensen (Aarhus)]

MSC 2000:: *11J83 Metric theory of numbers
11J13 Simultaneous homogeneous approximation, linear forms
11K60 Diophantine approximation (probabilistic number theory)

Keywords: Diophantine approximation; planar curves; Hausdorff measure

Citations: Zbl 1185.11047

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