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

TY - JOUR

T1 - Diophantine approximation on curves and the distribution of rational points

T2 - contributions to the divergence theory

AU - Beresnevich, Victor

AU - Vaughan, R.C.

AU - Velani, Sanju

AU - Zorin, Evgeniy

N1 - © 2021 Published by Elsevier Inc. This is an author-produced version of the published paper. Uploaded in accordance with the publisher’s self-archiving policy.

PY - 2021/9/17

Y1 - 2021/9/17

N2 - In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in $\mathbb{R}^n$. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For $n \geq 3$, the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarnik type theorem for divergence for arbitrary non-degenerate curves in $\mathbb{R}^n$.

AB - In this paper we develop an explicit method for studying the distribution of rational points near manifolds. As a consequence we obtain optimal lower bounds on the number of rational points of bounded height lying at a given distance from an arbitrary non-degenerate curve in $\mathbb{R}^n$. This generalises previous results for analytic non-degenerate curves. Furthermore, the main results are proved in the inhomogeneous setting. For $n \geq 3$, the inhomogeneous aspect is new even under the additional assumption of analyticity. Applications of the main distribution theorem also include the inhomogeneous Khintchine-Jarnik type theorem for divergence for arbitrary non-degenerate curves in $\mathbb{R}^n$.

U2 - 10.1016/j.aim.2021.107861

DO - 10.1016/j.aim.2021.107861

M3 - Article

SN - 0001-8708

VL - 388

JO - Advances in Mathematics

JF - Advances in Mathematics

M1 - 107861

ER -