Approximation by Egyptian Fractions and the Weak Greedy Algorithm

TL;DR

This paper introduces the weak greedy approximation algorithm for Egyptian fractions, analyzing its properties, the sequences it produces, and conditions for uniqueness and existence of such approximations.

Contribution

The paper presents a new algorithm (WGAA) for Egyptian fraction approximation, exploring its behavior, sequence properties, and conditions for uniqueness and realization.

Findings

WGAA produces sequences satisfying specific approximation bounds.

Existence of sequences depends on the growth of the sequence $(a_n)$.

Conditions under which the approximation sequences are unique are characterized.

Abstract

Let $0 < \theta \leqslant 1$. A sequence of positive integers $(b_n)_{n=1}^\infty$ is called a weak greedy approximation of $\theta$ if $\sum_{n=1}^{\infty}1/b_n = \theta$. We introduce the weak greedy approximation algorithm (WGAA), which, for each $\theta$, produces two sequences of positive integers $(a_n)$ and $(b_n)$ such that a) $\sum_{n=1}^\infty 1/b_n = \theta$; b) $1/a_{n+1} < \theta - \sum_{i=1}^{n}1/b_i < 1/(a_{n+1}-1)$ for all $n\geqslant 1$; c) there exists $t\geqslant 1$ such that $b_n/a_n \leqslant t$ infinitely often. We then investigate when a given weak greedy approximation $(b_n)$ can be produced by the WGAA. Furthermore, we show that for any non-decreasing $(a_n)$ with $a_1\geqslant 2$ and $a_n\rightarrow\infty$, there exist $\theta$ and $(b_n)$ such that a) and b) are satisfied; whether c) is also satisfied depends on the sequence $(a_n)$. Finally, we address the uniqueness of $\theta$ and $(b_n)$ and apply our framework to specific sequences.