Rational points near manifolds and metric Diophantine approximation

TY - JOUR

T1 - Rational points near manifolds and metric Diophantine approximation

AU - Beresnevich, Victor

PY - 2012/1

Y1 - 2012/1

N2 - This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of $R^n$. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifolds -- a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near non-degenerate manifolds in dimensions $n>2$ and show that they are ubiquitous (that is uniformly distributed).

AB - This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarnik type theorems for submanifolds of $R^n$. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindzuk. They have been settled for planar curves but remain open in higher dimensions. In this paper, Khintchine and Jarnik type divergence theorems are established for arbitrary analytic non-degenerate manifolds regardless of their dimension. The key to establishing these results is the study of the distribution of rational points near manifolds -- a very attractive topic in its own right. Here, for the first time, we obtain sharp lower bounds for the number of rational points near non-degenerate manifolds in dimensions $n>2$ and show that they are ubiquitous (that is uniformly distributed).

KW - Number Theory

KW - simultaneous Diophantine approximation on manifolds,

KW - metric theory

KW - Khintchine theorem

KW - Jarnik theorem

KW - Hausdorff dimension

KW - ubiquitous systems

KW - rational points near manifolds

UR - http://www.scopus.com/inward/record.url?scp=84857757740&partnerID=8YFLogxK

U2 - 10.4007/annals.2012.175.1.5

DO - 10.4007/annals.2012.175.1.5

M3 - Article

SN - 0003-486X

VL - 175

SP - 187

EP - 235

JO - Annals of Mathematics

JF - Annals of Mathematics

IS - 1

ER -