Journal of Integer Sequences, Vol. 15 (2012), Article 12.1.6

An Irrationality Measure for Regular Paperfolding Numbers


Michael Coons
Department of Pure Mathematics
University of Waterloo
200 University Avenue West
Waterloo, Ontario N2L 6P1
Canada

Paul Vrbik
Department of Computer Science
University of Western Ontario
1151 Richmond Street North
London, Ontario N6A 5B7
Canada

Abstract:

Let F(z) = Σn ≥ 1 fn zn be the generating series of the regular paperfolding sequence. For a real number α the irrationality exponent μ(α), of α, is defined as the supremum of the set of real numbers μ such that the inequality |α - p/q| < q-μ has infinitely many solutions (p,q) ∈ Z × N. In this paper, using a method introduced by Bugeaud, we prove that

μ(F(1/b)) ≤ 275331112987/137522851840 = 2.002075359 ...

for all integers b ≥ 2. This improves upon the previous bound of μ(F(1/b)) ≤ 5 given by Adamczewski and Rivoal.

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(Concerned with sequence A014577.)

Received September 14 2011; revised version received December 14 2011. Published in Journal of Integer Sequences, December 27 2011.

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