Approximation by Egyptian fractions and the weak greedy algorithm

Abstract

Let 0<θ⩽1. A sequence of positive integers (bn)n=1∞ is called a weak greedy approximation of θ if ∑n=1∞1/bn=θ. We introduce the weak greedy approximation algorithm (WGAA), which, for each θ, produces two sequences of positive integers (an) and (bn) such that(a) ∑n=1∞1/bn=θ;(b) 1/an+1<θ−∑i=1n1/bi<1/(an+1−1) for all n⩾1;(c) there exists t⩾1 such that bn/an⩽t infinitely often.We then investigate when a given weak greedy approximation (bn) can be produced by the WGAA. Furthermore, we show that for any non-decreasing (an) with a1⩾2 and an→∞, there exist θ and (bn) such that (a) and (b) are satisfied; whether (c) is also satisfied depends on the sequence (an). Finally, we address the uniqueness of θ and (bn) and apply our framework to specific sequences.