Rational points near manifolds and metric Diophantine approximation

Rational points near manifolds and metric Diophantine approximation

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarńık type theorems for submanifolds of Rn. These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V.G. Sprindžuk. They have been settled for planar curves but remain open in higher dimensions. In this...

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

On the Abc Conjecture and Diophantine Approximation by Rational Points

We show that an earlier conjecture of the author, on diophantine approximation of rational points on varieties, implies the “abc conjecture” of Masser and Oesterlé. In fact, a weak form of the former conjecture is sufficient, involving an extra hypothesis that the variety and divisor admit a faithful group action of a certain type. Analogues of this weaker conjecture are proved in the split fun...

Metric Diophantine Approximation: the Khintchine–groshev Theorem for Nondegenerate Manifolds

The main objective of this paper is to prove a Khintchine type theorem on divergence of linear Diophantine approximation on nondegenerate manifolds, which completes earlier results for convergence. 2000 Math. Subj. Class. Primary 11J83; Secondary 11K60.

Metric Diophantine Approximation and Probability

Let pn qn pn qn x denote the nth simple continued fraction convergent to an arbitrary irrational number x De ne the sequence of approximation constants n x q njx pn qnj It was conjectured by Lenstra that for almost all x lim n n jfj j n and j x zgj F z where F z z log if z and log z log z if z This was proved in BJW and extended in Nai to the same conclusion for kj x where kj is a sequence of p...