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Diophantine approximation on the parabola with non-monotonic approximation functions

Published online by Cambridge University Press:  15 January 2019

JING–JING HUANG*
Affiliation:
Department of Mathematics and Statistics, University of Nevada, Reno, 1664 N. Virginia St., Reno, NV 89557, U.S.A.

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.

Type
Research Article
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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