A determinantal approach to irrationality

Title: A determinantal approach to irrationality
Creator: Zudilin, Wadim
Relation: Funding BodyARCGrant NumberDP170100466
Relation: Constructive Approximation Vol. 45, Issue 2, p. 301-310
Publisher Link: http://dx.doi.org/10.1007/s00365-016-9333-7
Publisher: Springer
Resource Type: journal article
Date: 2017
Description: It is a classical fact that the irrationality of a number ξ∈R follows from the existence of a sequence p_n/q_n with integral p_n and q_n such that q_nξ−p_n≠0 for all n and q_nξ−p_n→0 as n→∞. In this paper, we give an extension of this criterion in the case when the sequence possesses an additional structure; in particular, the requirement q_nξ−p_n→0 is weakened. Some applications are given, including a new proof of the irrationality of π. Finally, we discuss analytical obstructions to extend the new irrationality criterion further and speculate about some mathematical constants whose irrationality is still to be established.
Subject: irrationality; rational approximation; π; Hankel determinant; Fekete–Chebyshev constant; transfinite diameter
Identifier: http://hdl.handle.net/1959.13/1387661
Identifier: uon:32649
Identifier: ISSN:0176-4276
Rights: This is a post-peer-review, pre-copyedit version of an article published in Constructive Approximation. The final authenticated version is available online at: http://dx.doi.org/10.1007/s00365-016-9333-7.
Language: eng
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