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Counting rational points near planar curves. (English) Zbl 1316.11065

V. Beresnevich et al. established in [Ann. Math. (2) 166, No. 2, 367–426 (2007; Zbl 1137.11048)] that any sufficiently smooth non-degenerate planar curve is of Khinchin type for divergence, while R. C. Vaughan and S. Velani [Invent. Math. 166, No. 1, 103–124 (2006; Zbl 1185.11047)] showed that such curves are also of Khinchin type for convergence. The proof of the convergence case relies on an upper bound on the rational points near the curve. In this paper the author provides an asymptotic formula for the number of rational points near a curve (see below). These results may lead to a quantitative version of the Khinchin-type theorem. The aim of the quantitative theorem would be to obtain a result similar to that of W. Schmidt [Can. J. Math. 12, 619–631 (1960; Zbl 0097.26205)], which was a sharpening of the classical Khinchin theorem.
A little more precisely, here is the main result of the paper under review: if \(f\) is a sufficiently smooth real function defined on the interval \([a,b]\), then the number of rational points with denominator no larger than \(Q\) that lie within a \(\delta\)-neighborhood of the graph of \(f\) is asymptotically equivalent to \((b-a)\delta Q^2\).

MSC:

11J83 Metric theory
11K60 Diophantine approximation in probabilistic number theory
11J13 Simultaneous homogeneous approximation, linear forms
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References:

[1] [1]V. Beresnevich, D. Dickinson, S. Velani, and R. C. Vaughan, Diophantine approximation on planar curves and the distribution of rational points, Ann. of Math. 166 (2007), 367–426. · Zbl 1137.11048
[2] [2]H. L. Montgomery, Ten Lectures on the Interface between Analytic Number Theory and Harmonic Analysis, CBMS Reg. Conf. Ser. Math. 84, Amer. Math. Soc., 1994.
[3] [3]W. Schmidt, A metrical theorem in diophantine approximation, Canad. J. Math. 12 (1960), 619–631. · Zbl 0097.26205
[4] [4]E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, 2nd ed., Oxford Univ. Press, 1986. · Zbl 0601.10026
[5] [5]R. C. Vaughan, The Hardy–Littlewood Method, 2nd ed., Cambridge Tracts in Math. 125, Cambridge Univ. Press, 1997. · Zbl 0868.11046
[6] [6]R. C. Vaughan and S. Velani, Diophantine approximation on planar curves: the convergence theory, Invent. Math. 166 (2006), 103–124. · Zbl 1185.11047
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