More or Less Uniform Convergence

TL;DR

This paper investigates a hierarchy of convergence notions for sequences, revealing their stratification by countable ordinals and establishing strictness of these hierarchies through model theoretic and proof theoretic perspectives.

Contribution

It introduces two stratified notions of metastable convergence, shows their equivalence to existing concepts, and demonstrates the strictness of the hierarchy with explicit examples.

Findings

Uniform metastable convergence is equivalent to some alpha-uniform convergence.

Abstract omega-uniform convergence corresponds to bounded oscillation.

The hierarchy of convergence notions is strict, with explicit examples for each level.

Abstract

Uniform metastable convergence is a weak form of uniform convergence for a family of sequences. In this paper we explore the way that metastable convergence stratifies into a family of notions indexed by countable ordinals. We give two versions of this stratified family, loosely speaking, they correspond to the model theoretic and proof theoretic perspectives. For the model theoretic version, which we call abstract alpha-uniform convergence, we show that uniform metastable convergence is equivalent to abstract $\alpha$-uniform convergence for some alpha, and that abstract omega-uniform convergence is equivalent to uniformly bounded oscillation of the family of sequences. The proof theoretic version, which we call concrete alpha-uniform convergence, is less canonical (it depends on a choice of ordinal notation), but appears naturally when "proof mining" convergence proofs to obtain quantitative bounds. We show that these hierarchies are strict by exhibiting a family of which is concretely alpha+1-uniformly convergent but not abstractly alpha-uniformly convergent for each alpha<omega_1.