Irrationality of the reciprocal sum of doubly exponential sequences
Junnosuke Koizumi
TL;DR
The paper investigates how the reciprocal sum of a doubly exponential integer sequence constrains its growth, proving a main result that nearly fixes each term given a tail bound on $a_n^2/a_{n+1}$ and a finite reciprocal sum $r = \sum 1/a_n$. This yields exact characterizations of the Sylvester sequence and Millin/Fibonacci-like sequences, linking their defining growth to their reciprocal sums. It also introduces the pseudo-greedy expansion of a real $r$ into unit fractions, defines the gap sequence, and shows a deep connection to Erdős-Graham type irrationality problems, including a reformulation that reduces the Erdős-Graham question to a conjecture about gap behavior. The work provides a structural framework for understanding Type 2 irrationality sequences and suggests a countable description of the exceptional $ ooted{ ext{α}}$ values for which $igl\langle α^{2^n} \bigr\rangle$ yields rational sums, with broader implications for irrationality criteria and related open problems.
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 $α> 1$, sequences asymptotic to $α^{2^n}$ have irrational reciprocal sums. Furthermore, our observations provide heuristic insight into an open problem by Erdős and Graham.