Underapproximation by Egyptian fractions

TL;DR

This paper investigates Egyptian fraction underapproximations of real numbers, analyzing the effectiveness of a greedy algorithm and identifying cases where it provides the best approximation or not.

Contribution

It introduces an infinite set of rationals where the greedy algorithm yields optimal underapproximations and studies cases where it does not.

Findings

Greedy algorithm often produces unique best underapproximations

An infinite set of rationals with optimal greedy underapproximations is constructed

Cases where the greedy algorithm is not the best underapproximation are characterized

Abstract

An increasing sequence $(x_i)_{i=1}^n$ of positive integers is an $n$-term Egyptian underapproximation of $\theta \in (0,1]$ if $\sum_{i=1}^n \frac{1}{x_i} < \theta$. A greedy algorithm constructs an $n$-term underapproximation of $\theta$. For some but not all numbers $\theta$, the greedy algorithm gives a unique best $n$-term underapproximation for all $n$. An infinite set of rational numbers is constructed for which the greedy underapproximations are best, and numbers for which the greedy algorithm is not best are also studied.