Non degenerate Birkhoff functions

TL;DR

This paper introduces a new framework using correspondences to automatically satisfy non-degeneracy conditions in Diophantine approximation on manifolds, simplifying analysis of well-approximated numbers.

Contribution

It proposes a novel setting with correspondences that inherently meet non-degeneracy conditions, improving upon previous function-based approaches.

Findings

Non-degeneracy conditions are automatically satisfied in the new setting.

Correspondences provide a better-behaved alternative to functions.

The approach simplifies the analysis of Diophantine approximation on manifolds.

Abstract

It is known after the works of Mahler, Kleinbock, Margulis, Sprind\v{z}uk and others that very well approximated numbers on a manifold form a zero measure set, assuming non-degeneracy conditions. These non-degeneracy conditions are, in many applications, difficult to check. We propose here a setting in which these are automatic. Rather than functions, we consider correspondences which have a better behaviour.