Quantum Kronecker fractions

TL;DR

This paper investigates Kronecker fractions, a class of rational numbers with maximal convergence radius in a quantum number framework, characterizing their continued fractions and exploring their properties and connections to topology.

Contribution

It characterizes Kronecker fractions with palindromic continued fractions and describes all such fractions with prime denominators and those with denominators under 5000.

Findings

Kronecker fractions have palindromic continued fractions.

All Kronecker fractions with prime denominators are described.

Numerous families of Kronecker fractions are identified.

Abstract

A few years ago Morier-Genoud and Ovsienko introduced an interesting quantization of the real numbers as certain power series in a quantization parameter $q.$ It is known now that the golden ratio has minimal radius among all these series. We study the rational numbers having maximal radius of convergence equal to 1, which we call Kronecker fractions. We prove that the corresponding continued fraction expansions must be palindromic and describe all Kronecker fractions with prime denominators. We found several infinite families of Kronecker fractions and all Kronecker fractions with denominator less than 5000. We also comment on the irrational case and on the relation with braids, rational knots and links.