Asymptotic Analysis of Infinite Decompositions of a Unit Fraction into Unit Fractions
Yuhi Kamio
TL;DR
Problem: Determine asymptotics for expressing the unit fraction $1$ as an infinite sum of unit fractions. Approach: extend the finite $1=\sum_{i=1}^n 1/a_i$ case to the infinite version by developing a generalized Sylvester sequence $s_i(n)$ with $s_1(n)=n+1$ and $s_{i+1}(n)=s_i(n)^2-s_i(n)+1$, and its limit constant $c_n=\lim_{i\to\infty} s_i(n)^{2^{-i}}$, with $\sqrt{n}<c_n<\sqrt{n+1}$. Contribution: prove that, for any nondecreasing $a_i$ with $\sum_{i=1}^\infty 1/a_i=1$ and some index where $a_i\neq s_i(n)$, we have $\liminf_{i\to\infty} a_i^{2^{-i}}<c_n$, solving the general problem via a Sou05-based method. Significance: links finite and infinite unit-fraction decompositions, providing explicit asymptotic bounds and a constructive comparison framework.
Abstract
Paul Erdős posed a problem on the asymptotic estimation of decomposing 1 into a sum of infinitely many unit fractions in \cite{Erd80}. We point out that this problem can be solved in the same way as the finite case, as shown in \cite{Sou05}.