Generalizing a conjecture of Erdős and Graham via best Egyptian underapproximations

Abstract

Let \(\{u_n\}_{n=1}^{\infty}\) be Sylvester’s sequence (sequence A000058 in the OEIS), and let \(a_1 < a_2 < \cdots\) be any other sequence of positive integers satisfying \(\sum_{i=1}^\infty \frac{1}{a_i} = 1\). Erdős and Graham conjectured that

$$ \liminf_{n\to\infty} a_n^{\frac{1}{2^n}} < \lim_{n\to\infty} u_n^{\frac{1}{2^n}} = c_0 = 1.264085\ldots.$$

This conjecture has recently been resolved constructively by Kamio and independently by the present authors. In this paper, we focus on a generalization of this conjecture using a non-constructive approach. Specifically, assuming the unproven claim of Erdős and Graham that every rational number admits an eventually greedy best Egyptian underapproximation, we establish a more general asymptotic inequality involving best Egyptian underapproximations.