On $q$-real and $q$-complex numbers

Abstract

In arXiv:1812.00170 and arXiv:1908.04365, Morier-Genoud and Ovsienko introduced $q$-rational numbers $[x]_q$, rational functions specializing to $x$ at $q=1$, and their extension to $q$-real numbers, Laurent series agreeing with $[x]_q$ for rational $x$. It was conjectured in arXiv:2102.00891 that for every real $x>1$, $[x]_q$ has positive radius of convergence, with optimal common radius $R_*=(3-\sqrt{5})/2$, attained at the golden ratio. We prove that $[x]_q$ converges to a nonvanishing holomorphic function for every real $x>1$ and $|q|<3-2\sqrt{2}$. The proof gives an expansion of $1/[x]_q$ as a $q$-adically convergent series of rational functions, converging absolutely and locally uniformly on an explicit region. It also defines a positive analytic function for $q\in(-R_*,1)$. Using arXiv:2405.15970, we further obtain convergence for $|q|<2-\sqrt{3}$. We compute $[x]_q$ explicitly for some transcendental $x$, including ${\rm cotan}(1)$ and $e$. We also establish sharp inequalities for numerators and denominators of $q$-rationals when $|q|=1$ and determine the closure of the set of $[x]_q$ when such $q$ is not a root of unity. Next, we show that coefficientwise reduction modulo every $m\ge2$ is injective on the Cantor line (the extended real line with doubled up rationals and the Cantor set topology); for $m=2$, it identifies the Cantor line with $\mathbb P^1(\mathbb F_2((q)))$. We describe the inverse map, extend rationality results modulo every prime, characterize quadratic series over $\mathbb F_2(q)$ corresponding to quadratic irrationals, derive criteria for eventual parity of coefficients, and compute the real numbers corresponding to $1+q^n$. Finally, we propose a definition of $q$-complex number $[\tau]_q$, a meromorphic function in $\tau\in\mathbb C_+$ expressed via hypergeometric functions evaluated at modular functions of $\tau$.