Diophantine Approximation on Manifolds and the Distribution of Rational Points: Contributions to the Convergence Theory

TY - JOUR

T1 - Diophantine Approximation on Manifolds and the Distribution of Rational Points: Contributions to the Convergence Theory

AU - Beresnevich, Victor

AU - Vaughan, R. C.

AU - Velani, Sanju

AU - Zorin, Evgeniy

N1 - © 2016, The Author(s).

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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 ⊂ 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.

AB - 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 ⊂ 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.

UR - http://www.scopus.com/inward/record.url?scp=85024472797&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnv389

DO - 10.1093/imrn/rnv389

M3 - Article

SN - 1687-0247

VL - 2017

SP - 2885

EP - 2908

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

IS - 10

ER -