2004/06/15 by Jonathan Sondow, Sondow, Jonathan · 1 citation
Mathematics · Physics and Astronomy · #11J82 #Advanced Mathematical Identities #Advanced Mathematical Theories and Applications #FOS: Mathematics #Iterative Methods for Nonlinear Equations #Number Theory (math.NT) #math.NT #msc:11J82
paper · pdf · doi:10.48550/arxiv.math/0406300
14 pages, presented in part at Journeés Arithmetiques XXIII in Graz
arxiv created 2004/06/15 · openalex publication_date 2004/06/15 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In math.NT/0307308 we defined the irrationality base of an irrational number and, assuming a stronger hypothesis than the irrationality of Euler's constant, gave a conditional upper bound on its irrationality base. Here we develop the general theory of the irrationality exponent and base, giving formulas and bounds for them using continued fractions and the Fibonacci sequence. A theorem of Jarnik on Diophantine approximation yields numbers with prescribed irrationality measure. By another method we explicitly construct series with prescribed irrationality base. Many examples are given.