Quantitative Khintchine in Simultaneous Approximation

TL;DR

This paper provides an effective version of a recent breakthrough in Khintchine's theorem for simultaneous Diophantine approximation on manifolds, advancing the understanding of approximation rates.

Contribution

It introduces an effective form of the Khintchine theorem on nondegenerate manifolds, building on recent theoretical advances.

Findings

Established an explicit approximation rate in Khintchine's theorem

Extended the theorem to a broader class of manifolds

Provided quantitative bounds for Diophantine approximation

Abstract

In a ground-breaking work \cite{BY}, Beresnevich and Yang recently proved Khintchine's theorem in simultaneous Diophantine approximation for nondegenerate manifolds resolving a long-standing problem in the theory of Diophantine approximation. In this paper, we prove an effective version of their result.