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Abstract
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 nondegenerate curve in $\mathbb{R}^n$. This generalises previous results for analytic nondegenerate 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 KhintchineJarnik type theorem for divergence for arbitrary nondegenerate curves in $\mathbb{R}^n$.
Original language  English 

Article number  107861 
Number of pages  33 
Journal  Advances in Mathematics 
Volume  388 
Early online date  12 Jul 2021 
DOIs  
Publication status  Published  17 Sept 2021 
Bibliographical note
© 2021 Published by Elsevier Inc. This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy.Projects
 1 Finished

Programme GrantNew Frameworks in metric Number Theory
1/06/12 → 30/11/18
Project: Research project (funded) › Research