Diophantine approximation in Banach spaces

TL;DR

This paper extends Diophantine approximation theory to infinite-dimensional Banach spaces, establishing general Dirichlet-type theorems and exploring the concept of optimality and its relation to badly approximable points.

Contribution

It introduces a framework for simultaneous Diophantine approximation in infinite-dimensional Banach spaces and analyzes the conditions for theorem optimality.

Findings

Dirichlet-type theorems are established in general Banach space settings

Optimality of approximation theorems is characterized and distinguished from the existence of badly approximable points

Theoretical framework broadens understanding of Diophantine approximation in infinite dimensions

Abstract

In this paper, we extend the theory of simultaneous Diophantine approximation to infinite dimensions. Moreover, we discuss Dirichlet-type theorems in a very general framework and define what it means for such a theorem to be optimal. We show that optimality is implied by but does not imply the existence of badly approximable points.