Simultaneous Khintchine theorem on manifolds in positive characteristics: convergence case

TL;DR

This paper proves the convergence part of Khintchine's theorem for analytic manifolds over local fields of positive characteristic, extending previous methods to non-monotonic approximation functions.

Contribution

It extends Khintchine's theorem to non-monotonic approximation functions on manifolds over positive characteristic fields, introducing new geometric results in the function field setting.

Findings

Proved convergence case of Khintchine's theorem for non-monotonic functions

Extended the function field by adjoining roots to handle non-monotonicity

Established new results in the geometry of numbers over function fields

Abstract

We prove the convergence case of Khintchine's theorem, with general approximation functions that are not necessarily monotonic, for analytic nonplanar manifolds over local fields of positive characteristic. Our approach is based on the method of counting rational points near manifolds developed by Beresnevich and Yang. To address the scenario where the given approximating function is not monotonic, we extend our function field by adjoining an appropriate root. Additionally, in the course of the proof, we establish several new results in the geometry of numbers over function fields, which are of independent interest.