On radius of convergence of $q$-deformed real numbers

TL;DR

This paper investigates the analytic properties and radius of convergence of $q$-deformed real numbers, proposing a conjecture that the $q$-deformed golden ratio has the smallest radius among all real numbers, supported by proofs and experiments.

Contribution

It introduces a conjecture relating the radius of convergence of $q$-deformed real numbers to classical irrational numbers, and provides proofs for specific cases along with computational evidence.

Findings

Conjecture that $q$-deformed golden ratio has minimal radius of convergence.

Proved the conjecture for certain rational numbers.

Established explicit lower bounds for ratios of golden and silver ratios.

Abstract

We study analytic properties of ``$q$-deformed real numbers'', a notion recently introduced by two of us. A $q$-deformed positive real number is a power series with integer coefficients in one formal variable~$q$. We study the radius of convergence of these power series assuming that $q$ is a complex variable. Our main conjecture, which can be viewed as a $q$-analogue of Hurwitz's Irrational Number Theorem, claims that the $q$-deformed golden ratio has the smallest radius of convergence among all real numbers. The conjecture is proved for certain class of rational numbers and confirmed by a number of computer experiments. We also prove the explicit lower bounds for the radius of convergence for the $q$-deformed convergents of golden and silver ratios.