Simultaneous Khintchine theorem on manifolds in positive characteristics: convergence case

TL;DR

This work establishes the convergence case of the Khintchine–Groshev theorem for analytic non-planar manifolds over local fields of positive characteristic, extending prior real-field results to function fields. The authors adapt the BY counting framework to the ultrametric function-field setting, confronting new challenges from extending to $\mathcal{K}_{\ell}$ and handling discrete subgroups; they develop auxiliary geometry-of-numbers results and a first-minima analysis to bound rational points near manifolds. The main outputs are a metric-zero result for almost every point on the manifold under the convergent Khintchine sum and a corresponding Hausdorff-dimension bound under joint summability conditions. These results deepen the understanding of Diophantine approximation on manifolds in positive characteristic and introduce techniques applicable to discrete subgroups in function-field geometry. The findings enhance the toolkit for metric Diophantine problems over function fields and link ultrametric analysis with lattice-point counting in non-archimedean settings.

Abstract

In this article, we prove the convergence case of Khintchine's theorem for analytic nonplanar manifolds over local fields of positive characteristic, in the setting of simultaneous Diophantine approximation. Our approach is based on the method of counting rational points near manifolds developed by Beresnevich and Yang. The results obtained here extend the work of Beresnevich and Yang, and Beresnevich and Datta, to the function field setting. In the course of the proof, we also establish several new results in the geometry of numbers over function fields, which are of independent interest.

Theorems & Definitions (36)

• Theorem 1.1: The Inhomogeneous Khintchine theorem
• Theorem 1.2
• Theorem 1.3
• Proposition 1
• Definition 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3: Duality