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

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.