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arXiv · 1809.06159

Diophantine approximation on curves and the distribution of rational points: divergence theory

V. Beresnevich·R. C. Vaughan·S. Velani·E. Zorin
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Abstract

In this paper we develop a new explicit method to studying rational points near manifolds and obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are also proved in the inhomogeneous setting. Applications of the main theorem include the Khintchine-Jarník type theorem for divergence for arbitrary non-degenerate curve in $\mathbb{R}^n$.

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V. Beresnevich, R. C. Vaughan, S. Velani, E. Zorin. 2018-09-17. Diophantine approximation on curves and the distribution of rational points: divergence theory. https://arxiv.org/abs/1809.06159

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