On eventually greedy best underapproximations by Egyptian fractions

Abstract

Erdős 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 "Erdős problems".]

7 pages; v2 corrects the history of the claim for rational numbers; v3 incorporates referee's suggestions. The author is grateful to Richard Green for discussing the paper and its background in the popular newsletter "A Piece of the Pi: mathematics explained"