Irrationality of fast converging series of rational numbers (Q2757137)

Call \(\sum_{n=0}^\infty u_n\) fast converging if \(|u_n|\leq Ch^{2^n}\) for some constants \(C>0\), \(0<h<1\). Some classical irrationality results involve series of this form, for example \(\sum_{n=0}^\infty{1\over 2^{2^n}+1}\), and more generally \(\sum_{n=0}^\infty{1\over a^{2^n}+b_n}\) provided \(\sum|b_na^{-2^n}|<\infty\) and \(a^{2^n}+b_n\neq 0\) for \(n\geq 0\). However, fast converging series can be rational, for example \(\sum_{n=0}^\infty{x^{2^n}\over 1-x^{2^{n+1}}}={x\over 1-x}\). Note also the quadratic irrational \(\sum_{n=0}^\infty{1\over f_{2^n}}={7-\sqrt 5\over 2}\), where \(f_n\) is the \(n\)-th Fibonacci number. These problems led Erdős to ask: Is it true that if \({u_{n+1}\over u_n^2}\rightarrow 1\) then \(\sum{1\over u_n}\) is irrational unless \(u_{n+1}=u_n^2-u_n+1\) for \(n\geq n_0\)?NEWLINENEWLINENEWLINEThe author considers fast converging series of type \(S=\sum_{n=0}^\infty{a_n\over b_nu_n}\) where \(a_n, b_n, u_n\) are integers, \(u_n\rightarrow\infty\), \(cu_n^2\leq u_{n+1}\leq c'u_n^2\) and \(a_n=O(u_n^\alpha)\) for some constants \(c,c'>0\), \(0<\alpha<1\) and with \(b_n=O(u_n^\varepsilon)\) for every \(\varepsilon>0\). The main result is that if \(\alpha<{1\over 7}\), then \(S\) is irrational unless NEWLINE\[NEWLINEu_{n+1}={p_n\over q_n}u_n^2-{a_{n+1}b_n\over a_nb_{n+1}}u_n+ {a_{n+2}b_{n+1}q_{n+1}\over a_{n+1}b_{n+2}p_{n+1}}NEWLINE\]NEWLINE for \(n\geq N(\alpha)\) where NEWLINE\[NEWLINEp_n=O(u_n^{\mu-2}u_{n+1}), q_n=O(u_n^\mu), \left|{u_{n+1}\over u_n^2}-{p_n\over q_n}\right|\leq{1\over q_nu_n^\mu}NEWLINE\]NEWLINE for every \(\mu\) with \(3\alpha<\mu<1-4\mu\). The proof follows a weak form of Mahler's method in transcendence theory. The author also obtains irrationality measures in some special cases for series of this form.