Interesting Esoterica

An Irrationality Measure for Regular Paperfolding Numbers

Article by Michael Coons and Paul Vrbik
  • Published in 2012
  • Added on
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Let $F(z) = \sum_{n \geq 1} f_n z^n$ be the generating series of the regular paperfolding sequence. For a real number $\alpha$ the irrationality exponent $\mu(\alpha)$, of $\alpha$, is defined as the supremum of the set of real numbers $\mu$ such that the inequality $\lvert \alpha - p/q \rvert \lt q-\mu$ has infinitely many solutions $(p,q) \in Z \times N$. In this paper, using a method introduced by Bugeaud, we prove that \[ \mu(F(1/b)) \leq 275331112987/137522851840 = 2.002075359 \ldots \] for all integers $b \geq 2$. This improves upon the previous bound of $\mu(F(1/b)) \leq 5$ given by Adamczewski and Rivoal.

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BibTeX entry

@article{AnIrrationalityMeasureforRegularPaperfoldingNumbers,
	title = {An Irrationality Measure for Regular Paperfolding Numbers},
	abstract = {Let {\$}F(z) = \sum{\_}{\{}n \geq 1{\}} f{\_}n z^n{\$} be the generating series of the regular paperfolding sequence. For a real number {\$}\alpha{\$} the irrationality exponent {\$}\mu(\alpha){\$}, of {\$}\alpha{\$}, is defined as the supremum of the set of real numbers {\$}\mu{\$} such that the inequality {\$}\lvert \alpha - p/q \rvert \lt q-\mu{\$} has infinitely many solutions {\$}(p,q) \in Z \times N{\$}. In this paper, using a method introduced by Bugeaud, we prove that

\[ \mu(F(1/b)) \leq 275331112987/137522851840 = 2.002075359 \ldots \]

for all integers {\$}b \geq 2{\$}. This improves upon the previous bound of {\$}\mu(F(1/b)) \leq 5{\$} given by Adamczewski and Rivoal.},
	url = {https://cs.uwaterloo.ca/journals/JIS/VOL15/Coons/coons3.html https://cs.uwaterloo.ca/journals/JIS/VOL15/Coons/coons3.pdf},
	author = {Michael Coons and Paul Vrbik},
	comment = {},
	urldate = {2016-07-11},
	collections = {Things to make and do},
	year = 2012
}