Approximating $1$ by Sums of Unit Fractions

Abstract

If a finite collection (with repetitions allowed) of unit fractions has sum larger than $K$, how close can a sum of some of them come to $1$ from below? Improving a bound of Erdős and Graham, we prove that the gap is at most $\exp (-c\sqrt{K\log K})$ for an absolute constant $c>0$ and all sufficiently large $K$. The proof combines equal fractions and applies a Fourier estimate to groups of denominators with comparable contributions. Elementary divisor counts then give the bound.

arXiv commentSubstantially simplified the v1