Approximation by Egyptian fractions and the weak greedy algorithm

Let $0<\theta ⩽1$. A sequence of positive integers ${\left(b_n\right)}_{n=1}^{\infty }$ 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 $\left(a_n\right)$ and $\left(b_n\right)$ 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 $n⩾1$;

(c) there exists $t⩾1$ such that $b_n/a_n⩽t$ infinitely often.

We then investigate when a given weak greedy approximation $\left(b_n\right)$ can be produced by the WGAA. Furthermore, we show that for any non-decreasing $\left(a_n\right)$ with $a_1⩾2$ and $a_n\to \infty$, there exist $\theta$ and $\left(b_n\right)$ such that (a) and (b) are satisfied; whether (c) is also satisfied depends on the sequence $\left(a_n\right)$. Finally, we address the uniqueness of $\theta$ and $\left(b_n\right)$ and apply our framework to specific sequences.