Irrationality of the reciprocal sum of doubly exponential sequences

Abstract

We show that sequences of positive integers whose ratios $a_n^2/a_{n+1}$ lie within a specific range are almost uniquely determined by their reciprocal sums. For instance, the Sylvester sequence is uniquely characterized as the only sequence with $a_n^2/a_{n+1}\in [2/3,4/3]$ whose reciprocal sum is equal to $1$. This result has applications to irrationality problems. We prove that for almost every real number $\alpha > 1$, sequences asymptotic to $\alpha^{2^n}$ have irrational reciprocal sums. Furthermore, our observations provide heuristic insight into an open problem by Erd\H{o}s and Graham.