Khintchine’s theorem and Diophantine approximation on manifolds

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.

Keywords

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

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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