2021/02/01 by Ludivine Leclere, Sophie Morier-Genoud, Leclere, Ludivine +5 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #Computability, Logic, AI Algorithms #FOS: Mathematics #Quantum Algebra (math.QA)
paper · pdf · doi:10.48550/arxiv.2102.00891
openalex publication_date 2021/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.