Doorn 2025 partitions prescribed sum reciprocals asymptotic bounds

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theorem_1: Bounds the least integer beyond which every integer has a partition into distinct parts with reciprocal sum α = p/q by a constant times q log³ q over min(α,1) log log q plus c(e+ε)^(2α).

theorem_2: Shows that every non-degenerate interval of non-negative reals contains, for each large enough prime q, a positive rational p/q such that no n up to 0.3 q log² q has a partition into distinct parts with reciprocal sum p/q.

theorem_3: Shows that for every fixed α > 0 the least integer beyond which every integer has a partition into distinct parts at least m with reciprocal sum α is (1/2 + o(1))(e^(2α) − 1)m² as m grows.

theorem_4: Bounds the number of positive rationals α whose α-partition threshold is at most n, and the number of reciprocal sums of distinct-part partitions of n, between exp((π/√3 − o(1))√(n/log n)) and exp((π/√3)√n).

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Wouter van Doorn, Partitions with prescribed sum of reciprocals: asymptotic bounds. arXiv:2502.02200 (2025).

The copy read for this card is arXiv:2502.02200v2 (23 July 2025; v1 4 February 2025), 12 pages; the arXiv listing read carries no journal reference and no journal record was found, so the paper is an author preprint. Read status: claims checked. The definitions of Section 1 (p. 1) and the statements of Theorems 1 (p. 2), 2 (p. 5), 3 (p. 6) and 4 (p. 11) were read clause by clause on the page images; no proof was checked. Result pages: Theorem 1, Theorem 2, Theorem 3 and Theorem 4. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2502.02200), every other right reserved.

Graham proved that every n >= 78 is a sum of distinct positive integers whose reciprocals sum to 1, and more generally that for each positive rational alpha and integer m there is a threshold n_{alpha,m} beyond which every n has an m-large alpha-partition, but his non-constructive input gave no bound on that threshold. Van Doorn supplies the first bounds. Theorem 1 shows that for every epsilon > 0 there is c with n_alpha < c q log^3 q / (min(alpha,1) log log q) + c(e + epsilon)^{2 alpha} for alpha = p/q, so n_alpha = o(q log^3 q) for alpha in (epsilon, 1/epsilon), and the lower bound of Theorem 2 shows n_alpha is not o(q log^2 q), which the paper reads as placing the upper bound within less than one log factor of optimal for alpha bounded away from 0; the proof combines results on unit fractions of Yokota, Bloom, and Liu and Sawhney with the author's earlier work, glued by lemmas that build alpha-partitions from M-free gamma-partitions. For larger m, Section 3 uses theorems of Croot to obtain the asymptotically optimal n_{alpha,m} = (1/2 + o_alpha(1))(e^{2 alpha} - 1) m^2 for fixed alpha > 0. Section 4 studies A(n), the set of alpha with n_alpha <= n, and proves |A(n)| = e^{n^{1/2 + o(1)}} by combining M-free partition results with known partition-function asymptotics. These quantitative bounds are the paper's contribution to problem 283 on partitions with prescribed reciprocal sum.

Source: https://arxiv.org/abs/2502.02200.

Bears on. #283 (whether, for a suitable polynomial $p$, every large integer is a sum of $p(n_i)$ over distinct $n_i$ with $\sum 1/n_i=1$): all four theorems concern the case $p(x)=x$ in the form with reciprocal sum any positive rational $\alpha$ in place of $1$. Theorem 1 bounds the threshold $n_\alpha$ from above, Theorem 2 shows that for suitable $\alpha=p/q$ it can exceed $0.3q\log^2q$, Theorem 3 gives its asymptotic size as $m\to\infty$ with $\alpha$ fixed when every part must be at least $m$, and Theorem 4 bounds the number of $\alpha$ with $n_\alpha\le n$. At reciprocal sum $1$, the problem's own case, Graham's $n_1=78$ is already exact (p. 1); the paper does not treat other polynomials.

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