Inhomogeneous theory of dual Diophantine approximation on manifolds

TL;DR

This paper develops an inhomogeneous Groshev type theory for dual Diophantine approximation on manifolds, introducing the concept of nice manifolds and generalizing existing homogeneous results, with applications to algebraic integer approximation.

Contribution

It introduces the notion of nice manifolds and establishes the divergence part of inhomogeneous dual Diophantine approximation theory for these manifolds, extending classical results.

Findings

Established divergence results for nice manifolds in inhomogeneous approximation

Generalized measure and dimension theorems to inhomogeneous settings

Applied multivariable theory to algebraic integer approximation problems

Abstract

The inhomogeneous Groshev type theory for dual Diophantine approximation on manifolds is developed. In particular, the notion of nice manifolds is introduced and the divergence part of the theory is established for all such manifolds. Our results naturally incorporate and generalize the homogeneous measure and dimension theorems for non-degenerate manifolds established to date. The generality of the inhomogeneous aspect considered within enables us to make a new contribution even to the classical theory in R^n. Furthermore, the multivariable aspect considered within has natural applications beyond the standard inhomogeneous theory such as to Diophantine problems related to approximation by algebraic integers.