Diophantine approximation on manifolds and the distribution of rational points: Contributions to the convergence theory

Victor Beresnevich, Robert C. Vaughan, Sanju Velani, Evgeniy Zorin

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Abstract

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 M ⊂ ℝn is of dimension strictly greater than (n+1)/2 and satisfies a natural non-degeneracy condition, then 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.

Original languageEnglish (US)
Pages (from-to)2885-2908
Number of pages24
JournalInternational Mathematics Research Notices
Volume2017
Issue number10
DOIs
StatePublished - May 1 2017

All Science Journal Classification (ASJC) codes

  • Mathematics(all)

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