Kamio 2025 asymptotic analysis infinite decompositions unit fraction

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theorem_8: States that any nondecreasing sequence of positive integers with reciprocal sum 1/n other than the generalized Sylvester sequence s_i(n) has liminf of a_i^(2^-i) below c_n = lim s_i(n)^(2^-i); n = 1 answers Problem 315.

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Yuhi Kamio, Asymptotic Analysis of Infinite Decompositions of a Unit Fraction into Unit Fractions. arXiv:2503.02317 (2025).

The copy read for this card is arXiv:2503.02317v1 (4 March 2025; the paper is dated 5 March 2025), 5 pages, the only version on the arXiv listing read; the listing carries no journal reference and no journal record was found (Crossref bibliographic query the same day), so the paper is an author preprint. Read status: claims checked. Problem 1, Definition 3, Proposition-Definition 4, Proposition 5 and Theorem 8 (pp. 1--3) were read clause by clause on the rendered page images of pp. 1 and 3 and the text layer of p. 2 on 2026-09-18, and rechecked on the page images of pp. 1--5 on 2026-10-08; the proof (pp. 3--5) was read for structure only and is not verified. Result page: theorem_8. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2503.02317), every other right reserved.

The paper settles a problem Erdos posed in Erdos-Graham (1980, p. 41) on the asymptotics of writing 1 as a sum of infinitely many unit fractions: if u_i is the Sylvester sequence 2, 3, 7, 43, ... (the paper's Problem 1 on p. 1 prints the recursion as u_0 = 1, u_{i+1} = u_i(u_i+1) + 1, which gives 1, 3, 13, ... and is not the sequence the theorem uses; its footnote says it corrects the monograph's u_{i+1} = u_i(u_i+1) after the site's page) and a_1 <= a_2 <= ... is any other sequence with sum of 1/a_i equal to 1, must liminf a_i^(2^{-i}) be strictly less than lim u_i^(2^{-i}) = 1.2640...? The author answers yes, and in the more general Theorem 8: for a positive integer n and the generalized Sylvester sequence s_1(n) = n+1, s_{i+1}(n) = s_i(n)^2 - s_i(n) + 1, if a_1 <= a_2 <= ... are positive integers with sum of 1/a_i equal to 1/n and a_i differs from s_i(n) for some i, then liminf a_i^(2^{-i}) < c_n, where c_n = lim s_i(n)^(2^{-i}) and sqrt(n) < c_n < sqrt(n+1) (Proposition-Definition 4). The method transfers Soundararajan's argument for the finite version of the problem (the bound a_n <= u_n - 1 for finite decompositions of 1) to the infinite case via product inequalities on partial products of the a_i, proved by induction with a shifting reduction to the case a_1 not equal to n+1. The case n = 1 answers yes to the question recorded as Problem 315, for every nondecreasing sequence and so for the strictly increasing ones the problem asks about.

Source: https://arxiv.org/abs/2503.02317.

Bears on.

• #315: Theorem 8 with n = 1 (p. 3) is the problem's question for nondecreasing sequences, answered yes; recorded on theorem_8.

Results to transcribe.

• Theorem 8 (p. 3; on the theorem_8 page): For a positive integer n, if positive integers a_1 <= a_2 <= ... satisfy sum 1/a_i = 1/n and a_i differs from s_i(n) for some i, then liminf a_i^(2^{-i}) < c_n = lim s_i(n)^(2^{-i}).
• The case n = 1 (the paper labels no corollary; p. 1 says it solves Problem 1 and its generalization, Theorem 8): any nondecreasing sequence of positive integers other than the Sylvester sequence s_i(1) = 2, 3, 7, 43, ... with sum of reciprocals 1 has liminf a_i^(2^{-i}) < c_1 = lim s_i(1)^(2^{-i}) = 1.2640..., answering Erdos's problem.
• Proposition-Definition 4 (p. 2; on the theorem_8 page): The limit c_n = lim s_i(n)^(2^{-i}) exists and satisfies sqrt(n) < c_n < sqrt(n+1), so c_n increases in n.
• Proposition 5 (p. 2; on the theorem_8 page): For positive integers n and j, the generalized Sylvester sequence satisfies sum_{i<j} 1/s_i(n) + 1/(s_j(n)-1) = 1/n, and s_j(n) - 1 is the product of the earlier terms times n.

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