Acta Mathematica

Volume 231 (2023)

Number 1

Khintchine’s theorem and Diophantine approximation on manifolds

Pages: 1 – 30



Victor Beresnevich (Department of Mathematics, University of York, Heslington, York, United Kingdom)

Lei Yang (College of Mathematics, Sichuan University, Chengdu, Sichuan, China)


In this paper we initiate a new approach to studying approximations by rational points to points on smooth submanifolds of $\mathbb{R}^n$. Our main result is a convergence Khintchine type theorem for arbitrary non-degenerate submanifolds of $\mathbb{R}^n$, which resolves a longstanding problem in the theory of Diophantine approximation. Furthermore, we refine this result using Hausdorff $s$-measures and consequently obtain the exact value of the Hausdorff dimension of $\tau$-well approximable points lying on any non-degenerate submanifold for a range of Diophantine exponents $\tau$ close to $1/n$. Our approach uses geometric and dynamical ideas together with a new technique of ‘generic and special parts’. In particular, we establish sharp upper bounds for the number of rational points of bounded height lying near the generic part of a non-degenerate manifold. In turn, we give an explicit exponentially small bound for the measure of the special part of the manifold. The latter uses a result of Bernik, Kleinbock and Margulis.


Diophantine approximation on manifolds, Khintchine theorem, Jarník theorem, Hausdorff dimension, rational points near manifolds, quantitative non-divergence, spectrum of Diophantine exponents

2010 Mathematics Subject Classification

11J13, 11J83, 11K55, 11K60

Dedicated to G. A. Margulis on the occasion of his 75th birthday.

Received 10 June 2021

Received revised 12 January 2023

Accepted 8 June 2023

Published 29 September 2023