Beurling moving averages and approximate homomorphisms

TL;DR

This paper extends the theory of Beurling regular variation by exploring moving averages and approximate homomorphisms, revealing an algebraic structure and dichotomy in behavior, with implications for quantifier weakening and limit processes.

Contribution

It introduces a new algebraic framework for Beurling regular variation using group structures and approximate homomorphisms, expanding the understanding of regular variation.

Findings

Established a dichotomy: systems are either very regular or very irregular.

Extended quantifier weakening in the context of regular variation.

Analyzed effects of using limsup and liminf instead of limits.

Abstract

The theory of regular variation, in its Karamata and Bojani\'c-Karamata/de Haan forms, is long established and makes essential use of homomorphisms. Both forms are subsumed within the recent theory of Beurling regular variation, developed further here, especially certain moving averages occurring there. Extensive use of group structures leads to an algebraicization not previously encountered here, and to the approximate homomorphisms of the title. Dichotomy results are obtained: things are either very nice or very nasty. Quantifier weakening is extended, and the degradation resulting from working with limsup and liminf, rather than assuming limits exist, is studied.