On a density conjecture about unit fractions

TL;DR

This paper proves that any subset of natural numbers with positive upper density contains a finite subset whose reciprocals sum to one, resolving a longstanding question by Erdős and Graham.

Contribution

It establishes the existence of finite reciprocal-sum subsets within dense sets of natural numbers, advancing understanding of unit fraction representations.

Findings

Any positive upper density set contains a finite subset with reciprocals summing to 1

Answers a question posed by Erdős and Graham

Advances the theory of unit fractions in dense sets

Abstract

We prove that any set $A\subset \mathbb{N}$ of positive upper density contains a finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$, answering a question of Erd\H{o}s and Graham.