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Bettin 2025 lower bound number egyptian fractions
theorem_1: The 2025 explicit lower bound for the number of distinct sums of distinct unit fractions with denominators at most N, with leading constant two log two and an iterated-logarithm product valid for each k of at least four for which the k-fold logarithm of N is at least three halves.
Sandro Bettin, Loïc Grenié, Giuseppe Molteni, Carlo Sanna, A lower bound for the number of Egyptian fractions. arXiv:2509.10030 (v1, 12 September 2025, the only arXiv version listed on 2026-09-18); Mathematics of Computation, published online 22 January 2026, DOI 10.1090/mcom/4190 (Crossref record read, with no volume or pages assigned yet). The copy read for this card is the arXiv v1 text (12 pages); the published text was not compared, and the locators on this card and its result page are v1 locators. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2509.10030), every other right reserved.
Let E_N be the set of rationals representable as sums of distinct unit fractions with denominators at most N, equivalently the set of subsums of the N-th harmonic sum, the empty sum included, so that |E_N| is the S(N) of problem 320. Theorem 1 (p. 2) gives an explicit lower bound: ln|E_N| >= 2 ln 2 * (N/ln N) times 1 when ln_2 N >= 1, times ln_3 N when ln_3 N >= 1, and times (1 - (3/2)/ln_k N) prod_{j=3}^{k} ln_j N for every integer k >= 4 with ln_k N >= 3/2; the abstract writes the last case as ln|E_N|/ln 2 >= (2 - 3/ln_k N)(N/ln N) prod_{j=3}^{k} ln_j N, the same bound, and the proof (p. 9) gives the constant 1.4 in place of 3/2. This improves Bleicher and Erdos, who proved the same shape of bound only under the stronger condition ln_k N >= k and with leading constant ln 2 instead of essentially 2 ln 2; because larger k are now admissible the improvement is in order of growth, not just the constant. The paper also gives methods to compute |E_N| exactly and tabulates it for N up to 154. It bears directly on problems 320 and 321: problem 320 asks for the number of distinct subsums of the N-th harmonic sum, and problem 321 for the size of the largest subset of {1, ..., N} whose subsets have pairwise distinct reciprocal sums. For problem 321 the relevant object is the set U of integers N whose reciprocal is not a {-1, 0, 1}-combination of the earlier reciprocals (Section 2): the proof bounds |U(N)| from below and applies Lemma 2, |E_N| >= 2^{|U(N)|}, and U(N) has all its subset reciprocal sums distinct, so the same bound divided by ln 2 is a lower bound for that problem's R(N); the deduction is written on the result page, not in the paper.
Read status: claims checked. Theorem 1 (p. 2, page image), the definition of E_N (p. 1) and Lemmas 1 and 2 (p. 3) were read clause by clause; the proof (Lemmas 3-10, pp. 3-9) was read for structure and is not verified here. Result page: theorem_1.
Source: https://arxiv.org/abs/2509.10030; published version https://doi.org/10.1090/mcom/4190.
Results.
- Theorem 1 (p. 2): explicit three-case lower bound for ln|E_N|, with leading factor 2 ln 2 and condition relaxed to ln_k N >= 3/2 for k >= 4.
- Comparison with Bleicher-Erdős: Bleicher and Erdős had alpha = ln 2 under ln_k N >= k; the relaxed condition admits larger k and so a faster-growing bound.
- Exact computation: Gives algorithms to compute |E_N| exactly and a table of values for N up to 154.
No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.