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Diophantine approximation on the parabola with non-monotonic approximation functions

Part of: Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  15 January 2019

JING–JING HUANG
JING–JING HUANG*
Affiliation:
Department of Mathematics and Statistics, University of Nevada, Reno, 1664 N. Virginia St., Reno, NV 89557, U.S.A.
*
e-mail: jingjingh@unr.edu
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Abstract

We show that the parabola is of strong Khintchine type for convergence, which is the first result of its kind for curves. Moreover, Jarník type theorems are established in both the simultaneous and the dual settings, without monotonicity on the approximation function. To achieve the above, we prove a new counting result for the number of rational points with fixed denominators lying close to the parabola, which uses Burgess’s bound on short character sums.

MSC classification

Primary: 11J83: Metric theory
Secondary: 11J13: Simultaneous homogeneous approximation, linear forms

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 168 , Issue 3 , May 2020 , pp. 535 - 542
Copyright
Copyright © Cambridge Philosophical Society 2019

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Footnotes

†

Supported by the UNR VPRI startup grant 1201-121-2479.

References

REFERENCES

Beresnevich, V.. Rational points near manifolds and metric Diophantine approximation. Ann. Math. (2) 175 (2012), no. 1, 187–235.CrossRefGoogle Scholar
Beresnevich, V.. On a theorem of V. Bernik in the metric theory of Diophantine approximation. Acta Arith. 117 (2005), no. 1, 71–80.CrossRefGoogle Scholar
Beresnevich, V., Dickinson, D. and Velani, S.. Diophantine approximation on planar curves and the distribution of rational points. With an Appendix II by Vaughan, R. C.. Ann. Math. (2) 166 (2007), no. 2, 367–426.Google Scholar
Beresnevich, V., Vaughan, R. C., Velani, S. and Zorin, E.. Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory. Int. Math. Res. Not. IMRN (2017), no. 10, 2885–2908.Google Scholar
Beresnevich, V. and Zorin, E.. Explicit bounds for rational points near planar curves and metric Diophantine approximation. Adv. Math. 225 (2010), no. 6, 3064–3087.CrossRefGoogle Scholar
Bernik, V. I.. The exact order of approximation of almost all points of a parabola. Mat. Zametki 26 (1979), no. 5, 657–665, 813.Google Scholar
Bernik, V. I. and Dodson, M. M.. Metric Diophantine approximation on manifolds. Cambridge Tracts in Mathematics, 137. (Cambridge University Press, Cambridge, 1999).CrossRefGoogle Scholar
Budarina, N. and Dickinson, D.. Inhomogeneous Diophantine approximation on integer polynomials with non-monotonic error function. Acta Arith. 160 (2013), no. 3, 243–257.CrossRefGoogle Scholar
Burgess, D. A.. On character sums and L-series. II. Proc. London Math. Soc. (3) 13 (1963), 524–536.CrossRefGoogle Scholar
Dodson, M. M., Rynne, B. P. and Vickers, J. A. G.. Khintchine-type theorems on manifolds. Acta Arith. 57 (1991), no. 2, 115–130.CrossRefGoogle Scholar
Huang, J.–J.. Rational points near planar curves and Diophantine approximation. Adv. Math. 274 (2015), 490–515.CrossRefGoogle Scholar
Huang, J.–J.. Hausdorff theory of dual approximation on planar curves. J. Reine Angew. Math. 740 (2018), 63–76.CrossRefGoogle Scholar
Huang, J.–J.. The density of rational points near hypersurfaces. Preprint available at arXiv:1711.01390 [math.NT].Google Scholar
Huang, J.–J.. Integral points close to a space curve. Preprint available at arXiv:1809.07796.Google Scholar
Hussain, M.. A Jarník type theorem for planar curves: everything about the parabola. Math. Proc. Camb. Phil. Soc. 159 (2015), no. 1, 47–60.CrossRefGoogle Scholar
Iwaniec, H. and Kowalski, E.. Analytic Number Theory. American Mathematical Society Colloquium Publications, 53 (American Mathematical Society, Providence, RI, 2004).Google Scholar
Kubilyus, I.. On the application of I. M. Vinogradov’s method to the solution of a problem of the metric theory of numbers. Dokl. Akad. Nauk SSSR (N.S.) 67 (1949), 783–786.Google Scholar
Simmons, D.. Some manifolds of Khinchin type for convergence. J. Théor. Nombres Bordeaux 30 (2018), no. 1, 175–193.CrossRefGoogle Scholar
Tenenbaum, G.. Introduction to analytic and probabilistic number theory. Third edition. Graduate Studies in Mathematics, 163 (American Mathematical Society, 2015).CrossRefGoogle Scholar
Vaughan, R. C. and Velani, S.. Diophantine approximation on planar curves: the convergence theory. Invent. Math. 166 (2006), no. 1, 103–124.CrossRefGoogle Scholar