Irrationality exponents of certain fast converging series of rational numbers (Q2040174)

The authors study \textit{fast converging series} \(S\) built from a sequence of rational numbers \(x_n\) all greater than \(1\),\(\{x_n\}_{n\geq 1}\) satisfying \(x_{n+1}\geq x_n^2\hbox{ for all sufficiently large }n,\) and numbers \(\varepsilon_n\in\{-1,1\}\) in the following way \[S=\sum_{n=1}^{\infty}\,\frac{\varepsilon_n}{x_n}.\tag{*}\] The main result gives under certain conditions the exact value of the irrationality exponent of \(S\). The irrationality exponent \(\mu(\alpha)\) of an irrational number \(\alpha\) is defined by the supremum of the set of numbers \(\mu\) for which \[\left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{\mu}}\] has infinitely many rational solutions \(p/q\) (every irrational \(\alpha\) has \(\mu(\alpha)\geq 2\) ; if \(\mu(\alpha)>2\), \(\alpha\) is irrational by Roth's theorem). Defining for a non-zero rational number \(x\) the quantity \(\hbox{den} x\) to be the smallest positive integer \(d\) for which \(dx\) is an integer, the main result then is Theorem 1. Let \(\{x_n\}_{n\geq 1}\) be a sequence of rational numbers all greater than \(1\) such that \[x_{n+1}\geq x_n^2\hbox{ for all sufficiently large }n,\] and let \(\varepsilon_n\in\{-1,1\}\). Define \[\delta_1=\hbox{den}\,x_1,\ \delta_{n+1}=\delta_n^2 \hbox{den}\,\left(\frac{x_{n+1}}{x_n^2}\right)\ (n\geq 1).\] Assume that \[\log{\delta_{n+1}}=o(\log{x_n})\] as \(n\rightarrow\infty\). Then the irrationality exponent of the number \(S\) defined in \((\ast)\) is \[\mu(S)=\limsup\,\frac{\log{x_{n+1}}}{\log{x_n}}.\] The layout of the paper is as follows: \S1. Introduction (\(2\) pages) \S2. Continued fraction expansion of the series (\(4\frac{1}{2}\) pages) \S3. Proof of Theorem 1 (\(4\) pages) \S4. Applications (\(4\) pages) These cover gap series, Engel series, Pierce series and Cahen's constant. References (\(13\) items)