Problem 355

Is there a lacunary sequence $A\subseteq \mathbb{N}$ (so that $A=\{a_1<a_2<\cdots\}$ and there exists some $\lambda>1$ such that $a_{n+1}/a_n\geq \lambda$ for all $n\geq 1$ ) such that

\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}

contains all rationals in some open interval?

Source: erdosproblems.com/355

Number theory
Unit fractions

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SolvedProved

An accepted solution exists. The statement is true.

Proved, with the answer yes. Van Doorn and Kovač's Theorem 1(a) (Acta Arith. 223 (2026), 275--295; refereed) constructs, for every $\lambda\in(1,2)$ , a $\lambda$ -lacunary sequence of positive integers whose finite reciprocal sums contain every rational in $[0,2]$ ; their Theorem 1(c) shows no $2$ -lacunary sequence, hence no sequence with $\lambda\ge2$ , does this. The Bleicher--Erdős conjecture is refuted. The site's label is PROVED (LEAN); the suffix is a catalog label whose scope is qualified under Formalization and the Lean label below, and no local kernel credit is claimed. The claim page is van Doorn and Kovač's theorem, from which the standing derives.

DoKo25: van Doorn, W. and Kovač, V., Lacunary sequences whose reciprocal sums represent all rational numbers in an interval. arXiv:2509.24971 (v1 29 September 2025; v2 5 October 2025; v3 3 December 2025, 17 pages); Acta Arith. 223 (2026), 275--295, DOI 10.4064/aa251001-13-1, published online 15 April 2026 (the arXiv journal reference and the Crossref record); the published text is not compared with the preprint. Theorem 1, p. 2; Theorem 2, pp. 2--3 of the preprint. Library home: Doorn and Kovač 2025; result pages Doorn and Kovač 2025, Theorem 1 and Doorn and Kovač 2025, Theorem 2.
BlEr76: Bleicher, M. N. and Erdős, P., Denominators of Egyptian fractions. J. Number Theory 8 (1976), 157--168; Conjecture 4, p. 167. Library home: Bleicher and Erdős 1976; result page Bleicher and Erdős 1976, Conjecture 4.
ErGr80: Erdős, P. and Graham, R. L., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathématique 28, Université de Genève (1980), p. 58. Library home: Erdős and Graham 1980.

Problem 355

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