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Claim. For every $\lambda\in(1,2)$ there is a sequence of positive integers $n_1<n_2<\cdots$ with $n_{i+1}/n_i\ge\lambda$ for every $i$ whose finite sums of distinct reciprocals $\sum_{i\in I}1/n_i$, $I$ finite, include every rational number in $[0,2]$, hence every rational in the open interval $(0,2)$; the sequence can be chosen with $n_{i+1}/n_i\to2$ and with every rational in $(0,2]$ represented by infinitely many finite subsets. No sequence with $n_{i+1}/n_i\ge2$ for every $i$ has finite reciprocal sums containing all rationals of a non-empty open interval, so the range $\lambda<2$ is the best possible. The first statement answers the question of Problem 355 in the affirmative and refutes the conjecture of Bleicher and Erdős (1976) that no lacunary sequence fills an interval of rationals, the conjecture recorded on conjecture_4. The result is Theorem 1, parts (a), (b) and (c), of the paper filed as doorn_2025_lacunary_sequences_whose_reciprocal_sums_represent, with result pages theorem_1 and theorem_2; the latter gives the exact least upper bound $R(\lambda)$ on the length of a rational interval a $\lambda$-lacunary sequence can fill.

Source. Wouter van Doorn and Vjekoslav Kovač, Lacunary sequences whose reciprocal sums represent all rational numbers in an interval, arXiv:2509.24971 (v1 29 September 2025, v3 3 December 2025); Acta Arithmetica 223 (2026), 275--295, DOI 10.4064/aa251001-13-1, published online 15 April 2026. Part (c) is the paper's Corollary 5, a short argument of the kind attributed to Kakeya; parts (a) and (b) rest on Proposition 8, a sufficient condition for the reciprocal sums to fill the rationals of $[0,\sum_i1/n_i)$, and on the divisor-chain construction of Section 4; the published text is not compared with the preprint. The problem's thread records the route to the paper: Kovač's guess of 20 August 2025 that the answer is yes, Kovač's construction over the following days of a sequence with ratios in $[3/2,2]$ filling the rationals of $[0,1)$, van Doorn's variants with lacunarity arbitrarily close to $2$, and the announcement of the paper on 14 September 2025.

Acceptance. The paper is published in Acta Arithmetica, a refereed journal, and its acknowledgments thank an anonymous referee; that is the refereed evidence. The site's curator, Thomas Bloom, credits the result to van Doorn and Kovač in the commentary of the problem page, which carries the label PROVED (LEAN) and was last edited 18 November 2025; the curator is independent of the authors, and that credit is the reviewed evidence.

Formalization link. The linked Lean file, in van Doorn's repository at the commit of 1 June 2026 that last changed it, describes itself as a formalization of the paper's main results obtained by Aristotle from Harmonic; it imports Mathlib, contains no sorry and no axiom declaration, proves Theorem_1 (parts (a) and (b) without the infinitely-many-representations clause), Theorem_2, Theorem_3 and Theorem_4, and deduces the formal-conjectures statement erdos_355 from Theorem_1 at $\lambda=3/2$ with the interval $(0,1)$. The thread's comments of 30 January and 13 March 2026 describe the route: a simplified version of the construction at $\lambda=1.01$, given to Gemini3 (as the thread names it) to make it easier to formalize and then formalized by Aristotle, later extended with Aristotle to versions of the paper's Theorems 1--3 and 12. It is the authors' own formalization and therefore a link on this page; it was not built or audited here, so it gives no formalized evidence, and the Lean suffix of the site's label PROVED (LEAN) is the catalog's label. The formal-conjectures statement file, whose formal_proof attribute points to the repository's unpinned main branch, is a statement and is not linked.