On a density conjecture about unit fractions (with an appendix by Thomas F. Bloom and Bhavik Mehta)

Thomas F. Bloom

doi:10.4171/jems/1456

JournalsjemsVol. 27, No. 11pp. 4563–4589

On a density conjecture about unit fractions (with an appendix by Thomas F. Bloom and Bhavik Mehta)

Thomas F. Bloom
University of Oxford, UK
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Abstract

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

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Thomas F. Bloom, On a density conjecture about unit fractions (with an appendix by Thomas F. Bloom and Bhavik Mehta). J. Eur. Math. Soc. 27 (2025), no. 11, pp. 4563–4589

DOI 10.4171/JEMS/1456