$q$-deformed rationals and irrationals
Sophie Morier-Genoud, Valentin Ovsienko
TL;DR
The paper develops a coherent theory of $q$-deformed rational and irrational numbers by embedding them in a $\,PSL(2,\mathbb{Z})$-equivariant framework. It constructs explicit $q$-rationals via continued fractions and a $q$-deformed Farey graph, and proves structural properties such as total positivity, unimodality, and palindromicity of associated polynomials. It extends these ideas to $q$-irrationals through a stabilization mechanism, obtaining closed forms for quadratic cases and exploring higher-degree phenomena with conjectures on radii of convergence. The work also reveals surprising symmetries, left-right duals, and links to classical combinatorial sequences, highlighting both rich mathematical structure and numerous open questions.
Abstract
The concept of $q$-deformation, or ``$q$-analogue'' arises in many areas of mathematics. In algebra and representation theory, it is the origin of quantum groups; $q$-deformations are important for knot invariants, combinatorial enumeration, discrete geometry, analysis, and many other parts of mathematics. In mathematical physics, $q$-deformations are often understood as ``quantizations''. The recently introduced notion of a $q$-deformed real number is based on the geometric idea of invariance by a modular group action. The goal of this lecture is to explain what is a $q$-rational and a $q$-irrational, demonstrate beautiful properties of these objects, and describe their relations to many different areas. We also tried to describe some applications of $q$-numbers.