\documentclass[12pt,reqno]{article} \usepackage[usenames]{color} \usepackage{amssymb} \usepackage{graphicx} \usepackage{amscd} \usepackage[colorlinks=true, linkcolor=webgreen, filecolor=webbrown, citecolor=webgreen]{hyperref} \definecolor{webgreen}{rgb}{0,.5,0} \definecolor{webbrown}{rgb}{.6,0,0} \usepackage{color} \usepackage{fullpage} \usepackage{float} %\usepackage{psfig} \usepackage{graphics,amsmath,amssymb} \usepackage{amsthm} \usepackage{amsfonts} \usepackage{latexsym} \usepackage{epsf} \usepackage{listings} \setlength{\textwidth}{6.5in} \setlength{\oddsidemargin}{.1in} \setlength{\evensidemargin}{.1in} \setlength{\topmargin}{-.5in} \setlength{\textheight}{8.9in} \newcommand{\seqnum}[1]{\href{http://oeis.org/#1}{\underline{#1}}} \theoremstyle{plain} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{question}[theorem]{Question} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} % Put your definitions, symbols, abbreviations here. %\newcommand{}{} \begin{document} \begin{center} \epsfxsize=4in \leavevmode\epsffile{logo129.eps} \end{center} \begin{center} \vskip 1cm{\LARGE\bf An Irrationality Measure for Regular \\ \vskip .1in Paperfolding Numbers } \vskip 1cm \large Michael Coons\\ Department of Pure Mathematics\\ University of Waterloo\\ 200 University Avenue West\\ Waterloo, Ontario N2L 6P1\\ Canada\\ \href{mailto:mcoons@math.uwaterloo.ca}{\tt mcoons@math.uwaterloo.ca}\\ \ \\ Paul Vrbik\\ Department of Computer Science\\ University of Western Ontario\\ 1151 Richmond Street North\\ London, Ontario N6A 5B7\\ Canada\\ \href{mailto:pvrbik@csd.uwo.ca}{\tt pvrbik@csd.uwo.ca}\\ \end{center} \vskip .2 in \begin{center} {\em In memory of Alf van der Poorten} \end{center} \begin{abstract} Let $\mathbf{F}(z)=\sum_{n\geqslant 1}f_nz^n$ 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 $|\alpha-p/q|0$, we have for large enough $m$, say $m\geqslant m_0(k)$, that $$(1-\varepsilon)h_kb^{-2^{m}(2k+1)}\leqslant \left|\mathbf{F}(1/b)-\frac{P_{k,m}(1/b)}{Q_{k,m}(1/b)}\right|\leqslant (1+\varepsilon)h_kb^{-2^{m}(2k+1)}.$$ It remains to determine the degree of $P_{k,m}(z)$ and $Q_{k,m}(z)$. To this end, note that writing $$\frac{P_m(z)}{Q_m(z)}=\sum_{j=0}^{m-1}\frac{z^{2^{j}}}{1-z^{2^{j+2}}},$$ we have that $$\deg Q_m(z)\leqslant\deg (1-z^{2^{m+1}})=2^{m+1}$$ and that \begin{align*} \deg P_m(z) &\leqslant \max_{0\leqslant j\leqslant m-1}\left\{\frac{z^{2^{j}}(1-z^{2^{m+1}})}{{1-z^{2^{j+2}}}}\right\}\\ &= \max_{0\leqslant j\leqslant m-1}\{2^{m+1}-2^{j+2}-2^j\}\\ &=2^{m+1}-5\\ &\leqslant 2^{m+1}.\end{align*} Using the definitions of $P_m(z)$ and $Q_m(z)$, $$\deg Q_{k,m}(z)=\deg Q_m(z)Q_{k,0}(z^{2^m})=\deg Q_m(z)+\deg Q_{k,0}(z^{2^m})<2^{m}(k+2),$$ and $$\deg P_{k,m}(z)=\max\{\deg P_m(z)Q_{k,0}(z^{2^m}),P_{k,0}(z^{2^m})\}<2^m(k+2).$$ We continue following Bugeaud \cite{Bpreprint} by defining the integers $$p_{k,m}:=b^{2^m(k+2)}P_{k,m}(1/b)$$ and $$q_{k,m}:=b^{2^m(k+2)}Q_{k,m}(1/b).$$ Since $h_k$ is nonzero there exist positive real constants $c_i(k)$ ($i=1,\ldots,4$) depending only on $k$ so that \begin{equation}\label{qkm}c_1(k)b^{2^m(k+2)}\leqslant q_{k,m}\leqslant c_2(k)b^{2^m(k+2)},\end{equation} and \begin{equation}\label{Gpq}\frac{c_3(k)}{b^{2^{m+1}(k+1/2)}}\leqslant\left|\mathbf{F}(1/b)-\frac{p_{k,m}}{q_{k,m}}\right|\leqslant\frac{c_4(k)}{b^{2^{m+1}(k+1/2)}}.\end{equation} Note that $$2^{m}(k+2)=2^m(k+1/2)\left(\frac{k+2}{k+1/2}\right),$$ so that by \eqref{qkm} there are positive constants $c_5(k)$ and $c_6(k)$ such that $$\frac{c_5(k)}{q_{k,m}^{2\cdot\frac{k+1/2}{k+2}}}\leqslant \frac{1}{b^{2^{m+1}(k+1/2)}}\leqslant \frac{c_6(k)}{q_{k,m}^{2\cdot\frac{k+1/2}{k+2}}}.$$ Applying this to \eqref{Gpq} yields \begin{equation}\label{q2Gq2}\frac{c_3(k)c_5(k)}{q_{k,m}^{1+\frac{k-1}{k+2}}}\leqslant \left|\mathbf{F}(1/b)-\frac{p_{k,m}}{q_{k,m}}\right|\leqslant\frac{c_4(k)c_6(k)}{q_{k,m}^{1+\frac{k-1}{k+2}}}.\end{equation} Let $K\geqslant 3$ be an integer and denote by $m_1(k)$ an integer such that the sequence $\{q_{k-2,m}\}_{m\geqslant m_1(k)}$ is increasing. We define the sequence of positive integers $\{Q_{K,n}\}_{n\geqslant 1}$ as the sequence of all the integers $q_{k-2,m}$ with $k$ odd, $2^{K-1}-1\leqslant k\leqslant 2^K-1$, $m\geqslant m_1(k)$, put in increasing order. Thus by \eqref{qkm}, \eqref{q2Gq2}, and Lemma \ref{Blem2}, there exist positive constants $C_1(K),$ $C_2(K)$ and $C_3(K)$ such that both \begin{equation*}\label{need1} Q_{K,n}