Diophantine Approximation on Manifolds and the Distribution of Rational Points: Contributions to the Convergence Theory. (14th June 2016)

• Record Type: Journal Article
• Authors: Beresnevich, Victor; Vaughan, Robert C.; Velani, Sanju; Zorin, Evgeniy
• Is Part Of: International mathematics research notices. Volume 2017:Number 10(2017)
• Page Start: 2885
• Page End: 2908
• Publication Date: 2016-06-14
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1093/imrn/rnv389
• Languages: English
• ISSNs: 1073-7928

Abstract

Abstract : In memory of Klaus Roth (October 29, 1925 – November 10, 2015). In this article, we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $\mathcal{M} \subset \mathbb{R}^n$ is of dimension strictly greater than $(n+1)/2$ and satisfies a natural non-degeneracy condition, then $\mathcal{M}$ is of Khintchine type for convergence. The key lies in obtaining essentially the best possible upper bound regarding the distribution of rational points near manifolds.