Approximation by Egyptian Fractions and the Weak Greedy Algorithm

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.