Korsky 2026 stretched exponential bound erdos graham unit

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theorem_1_1: States the preprint's claim that every finite multiset of positive integers with reciprocal sum above K has a submultiset whose reciprocal sum lies in [1 - exp(-c sqrt(K log K)), 1], together with the barrier construction.

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Samuel Korsky, A Stretched-Exponential Bound for an Erdos-Graham Unit-Fraction Problem. arXiv preprint (2026). arXiv:2607.04157.

The retained folder-name PDF is arXiv:2607.04157v1 (5 July 2026; the title page is dated 7 July 2026), 27 pages, the only arXiv version on 2026-09-18. No journal record exists (arXiv listing and a Crossref title query, 2026-09-18), and in the site's discussion thread for Problem 312 the author wrote on 9 July 2026 that they have no plans to send this paper to a journal. Its text layer is clean, and the statements below were checked in it against the PDF pages. The arXiv record (https://arxiv.org/abs/2607.04157, read 2026-09-17) names the Creative Commons Attribution 4.0 license.

Read status: claims checked. Theorem 1.1, display (1.4) and Lemmas 2.1 and 2.2 were read clause by clause and the proofs of the two lemmas (p. 3) were read; Sections 3--4 and Appendices A--B (pp. 4--26) were read for structure only and are not verified here. A result page exists for Theorem 1.1, the statement Problem 312 consumes.

For a finite multiset A of positive integers with R(A) = sum 1/a, let eps(A) be 1 minus the largest reciprocal subsum of A that is at most 1. Theorem 1.1 proves eps(A) <= exp(-c sqrt(K log K)) whenever R(A) > K >= K_0, improving the Erdos-Graham bound eps(A) << K^{-2} toward the bound exp(-cK) that they asked about. The proof compresses the complement of a maximal subsum (Lemma 2.1, Lemma 2.2) into a stable multiset C whose multiplicities satisfy m_n < P^-(n) and no subsum of which lands in (1-N^{-2},1); a sparse activation lemma, with a divisor-sorting bound on high frequencies and a one-sided local limit lemma, shows that a random subsum would land there unless the mass is small, yielding R(C) << x^2/log x with x = log(1/eps(A)) and hence x >> sqrt(K log K). The paper also records the standard construction with p-1 copies of each prime p <= z, showing eps >= exp(-(1+o(1)) R log R), so the truth lies between the new bound and that barrier. On problem 312 the claimed bound has the exp(-c sqrt(K log K)) form, and the abstract, Theorem 1.1 and the proof outline are internally consistent as read. The paper's closing acknowledgment (p. 26) declares that a language model assisted the author extensively, both in filling in technical details (parts of the Fourier-analytic activation argument, the divisor-sorting analysis, the local-limit step, the divisor-incidence and crossing-sum estimates) and in drafting and revising the manuscript, while the conceptual reductions and proof strategy are described as the author's and the author assumes responsibility for the final form. In the site's discussion thread for Problem 312 (comments of 9 July 2026) a commenter questioned the paper's terminology as machine-written and its crediting of sources, and the author replied that the paper is their most AI-involved, that they reviewed it personally, that the section names were machine-generated, and that they do not plan to submit it to a journal. Those concerns are the thread's, not findings of this reading, which did not audit the paper's use of its sources; Section 1.1 does credit Croot, Bloom and Liu--Sawhney for the related exact-representation results, and the reference list names them with the 1980 monograph and Iwaniec--Kowalski.

Source: https://arxiv.org/abs/2607.04157.

Bears on. #312: Theorem 1.1 is the best upper bound found for the deficit eps(A), as an unrefereed preprint claim with declared AI assistance; the site's page does not cite the paper (as of the refresh) and the exp(-cK) bound that Erdos and Graham asked about remains open; display (1.4) is the barrier below which no uniform bound can go.

Results to transcribe.

• Theorem 1.1: There are absolute c > 0 and K_0 such that every finite multiset A with R(A) > K >= K_0 satisfies eps(A) <= exp(-c sqrt(K log K)).
• Lemma 2.1: Optimal complement: removing a maximal reciprocal subsum S leaves B = A \ S with R(B) > R(A) - 1 and every denominator of B smaller than N = 1/eps(A).
• Lemma 2.2: Stable compression: replacing p copies of 1/n by one copy of 1/(n/p) terminates in a multiset C with the same reciprocal mass, multiplicities m_n < P^-(n), and all subsums lifting to genuine subsums of A.
• Equation (1.4): The lower-bound construction with p-1 copies of each prime p <= z gives eps(A_z) >= exp(-(1+o(1)) R(A_z) log R(A_z)), bounding what any method can achieve.