Quantitative non-divergence and Diophantine approximation on manifolds

TL;DR

This survey discusses the role of Quantitative non-Divergence estimates in Diophantine approximation on manifolds, covering extremal manifolds, Khintchine-Groshev theorems, and rational points proximity.

Contribution

It highlights the applications of Quantitative non-Divergence estimates in Diophantine approximation, emphasizing their importance in understanding manifold approximation properties.

Findings

Application of non-Divergence estimates to extremal manifolds

Results on Khintchine-Groshev type theorems for manifolds

Insights into rational points near manifolds

Abstract

The goal of this survey is to discuss the Quantitative non-Divergence estimate on the space of lattices and present a selection of its applications. The topics covered include extremal manifolds, Khintchine-Groshev type theorems, rational points lying close to manifolds and badly approximable points on manifolds. The main emphasis is on the role of the Quantitative non-Divergence estimate in the aforementioned topics within the theory of Diophantine approximation, and therefore this paper should not be regarded as a comprehensive overview of the area.