Moscow Mathematical Journal

Volume 24, Issue 1, January–March 2024 pp. 1–19.

On Radius of Convergence of $q$-Deformed Real Numbers

Authors: Ludivine Leclere (1), Sophie Morier-Genoud (2), Valentin Ovsienko (3), and Alexander Veselov (4)
Author institution:(1) Laboratoire de Mathématiques, Université de Reims, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse – BP 1039, 51687 Reims cedex 2, France
(2) Laboratoire de Mathématiques, Université de Reims, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse – BP 1039, 51687 Reims cedex 2, France
(3) Centre National de la Recherche Scientifique, Laboratoire de Mathématiques, Université de Reims, U.F.R. Sciences Exactes et Naturelles, Moulin de la Housse – BP 1039, 51687 Reims cedex 2, France
(4) Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK

Summary:

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.

2020 Math. Subj. Class. 11A55, 05A30, 30B10.

Keywords: $q$-deformations, modular group, radius of convergence.

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