An Inhomogeneous Transference Principle and Diophantine Approximation

TL;DR

This paper introduces a new transfer principle that extends homogeneous Diophantine approximation results to the inhomogeneous setting, establishing the inhomogeneous Baker-Sprindzuk conjecture and related extremality results.

Contribution

It develops a novel approach to transfer homogeneous Diophantine approximation results to inhomogeneous cases, filling a significant gap in the theory.

Findings

Proved the inhomogeneous analogue of the Baker-Sprindzuk conjecture.

Established a complete inhomogeneous version of the extremality of friendly measures.

First step towards a unified inhomogeneous Diophantine approximation theory.

Abstract

In a landmark paper, D.Y. Kleinbock and G.A. Margulis established the fundamental Baker-Sprindzuk conjecture on homogeneous Diophantine approximation on manifolds. Subsequently, there has been dramatic progress in this area of research. However, the techniques developed to date do not seem to be applicable to inhomogeneous approximation. Consequently, the theory of inhomogeneous Diophantine approximation on manifolds remains essentially non-existent. In this paper we develop an approach that enables us to transfer homogeneous statements to inhomogeneous ones. This is rather surprising as the inhomogeneous theory contains the homogeneous theory and so is more general. As a consequence, we establish the inhomogeneous analogue of the Baker-Sprindzuk conjecture. Furthermore, we prove a complete inhomogeneous version of the profound theorem of Kleinbock, Lindenstrauss & Weiss on the extremality of friendly measures. The results obtained in this paper constitute the first step towards developing a coherent inhomogeneous theory for manifolds in line with the homogeneous theory.