PreprintCC BY 4.0arXiv
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√(Klog 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.
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BibTeX
@misc{korsky2026approximating,
title = {{Approximating $1$ by Sums of Unit Fractions}},
author = {Samuel Korsky},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2607.04157},
eprint = {2607.04157},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.04157},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Approximating $1$ by Sums of Unit Fractions
AU - Samuel Korsky
PY - 2026
DA - 2026/10/08
PB - arXiv
DO - 10.48550/arxiv.2607.04157
UR - https://arxiv.org/abs/2607.04157
AB - 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√(Klog 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.
ER -