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arXiv · 1506.09049

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 paper we develop the convergence theory of simultaneous, inhomogeneous Diophantine approximation on manifolds. A consequence of our main result is that if the manifold $M \subset \mathbb{R}^n$ 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.

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Victor Beresnevich, Robert C. Vaughan, Sanju Velani, Evgeniy Zorin. 2015-06-30. Diophantine approximation on manifolds and the distribution of rational points: contributions to the convergence theory. https://arxiv.org/abs/1506.09049

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