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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: [email protected]
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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
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

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