On eventually greedy best underapproximations by Egyptian fractions

TL;DR

This paper proves that for almost all positive real numbers, the greedy method does not produce the best Egyptian fraction underapproximations as the number of terms increases, resolving a problem posed by Erdős and Graham.

Contribution

It demonstrates that the set of numbers for which greedy Egyptian fraction underapproximations are optimal has Lebesgue measure zero, disproving a conjecture by Erdős and Graham.

Findings

The set of real numbers with eventually greedy best Egyptian underapproximations has measure zero.

The result applies to almost all positive real numbers, except a measure-zero set.

The paper resolves Problem 206 from Bloom's website.

Abstract

Erd\H{o}s and Graham found it conceivable that the best $n$-term Egyptian underapproximation of almost every positive number for sufficiently large $n$ gets constructed in a greedy manner, i.e., from the best $(n-1)$-term Egyptian underapproximation. We show that the opposite is true: the set of real numbers with this property has Lebesgue measure zero. [This note solves Problem 206 on Bloom's website "Erd\H{o}s problems".]