Harmonic Analysis on the Positive Rationals. Determination of the Group Generated by the Ratios $(an+b)/(An+B)$

TL;DR

This paper develops a method to determine the group generated by specific rational ratios using harmonic analysis and duality principles, providing a new approach to understanding their algebraic structure.

Contribution

It introduces a novel harmonic analysis technique to identify the group generated by ratios of the form (an+b)/(An+B) via duality, advancing the algebraic understanding of these ratios.

Findings

Successfully characterizes the group generated by the ratios (an+b)/(An+B).

Provides a duality-based method for analyzing rational ratios.

Enhances the theoretical framework for harmonic analysis on positive rationals.

Abstract

The authors show how to determine the group generated by the ratios $(an+b)/(An+B)$ through its dual.