Vol. 11, No. 2, 2022
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Improved constants for effective irrationality measures from hypergeometric functions

Paul M. Voutier
Vol. 11 (2022), No. 2, 161–180
DOI: 10.2140/moscow.2022.11.161
Abstract

We simplify and improve the constant c that appears in effective irrationality measures,

 |(a /b)m∕n − p /q| > c |q |−(κ+1),

obtained from the hypergeometric method for a∕b near 1. The dependence of c on |a| in our result is best possible (as is the dependence on n in many cases). For some applications, the dependence of this constant on |a| becomes important. We also establish some new inequalities for hypergeometric functions that are useful in other diophantine settings.

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Keywords
diophantine approximation, effective irrationality measures, hypergeometric functions
Mathematical Subject Classification
Primary: 11J82, 11J68, 33C05
Supplementary material

Appendix: Values of C_n, D_n and supporting data
Milestones
Received: 31 October 2021
Revised: 18 May 2022
Accepted: 1 June 2022
Published: 13 August 2022
Authors
Paul M. Voutier
London
United Kingdom