An approach to harmonic analysis on non-locally compact groups I: level structures over locally compact groups

TL;DR

This paper introduces a new class of spaces extending compactness concepts, enabling harmonic analysis on non-locally compact groups, with foundational topological results and potential applications in higher-dimensional number theory.

Contribution

It defines a novel framework for level structures over locally compact groups, generalizing compactness and establishing key topological theorems applicable to harmonic analysis.

Findings

Established analogues of Tychonoff's Theorem for the new spaces

Proposed applications in harmonic analysis on non-locally compact groups

Provided a foundation for further research in higher-dimensional number theory

Abstract

We define a class of spaces on which one may generalise the notion of compactness following motivating examples from higher-dimensional number theory. We establish analogues of several well-known topological results (such as Tychonoff's Theorem) for such spaces. We also discuss several possible applications of this framework, including the theory of harmonic analysis on non-locally compact groups.