Erdős Problem 47

References:

• erdosproblems.com/47
• [Er80] Erdős, Paul, A survey of problems in combinatorial number theory. Ann. Discrete Math. (1980), 89-115.
• [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).
• [Er92c] Erdős, P., Some of my forgotten problems in number theory. Hardy-Ramanujan J. (1992), 34-50.
• [Er95] Erdős, Paul, Some of my favourite problems in number theory, combinatorics, and geometry. Resenhas (1995), 165-186.
• [Er96b] Erdős, Paul, Some problems I presented or planned to present in my short talk. Analytic number theory, Vol. 1 (Allerton Park, IL, 1995) (1996), 333-335.
• [Er97c] Erdős, Paul, Some of my favorite problems and results. The mathematics of Paul Erdős, I (1997), 47-67.
• [Bl21] Bloom, T. F., On a density conjecture about unit fractions. arXiv:2112.03726 (2021).
• [LiSa24] Liu, Y. and Sawhney, M., On further questions regarding unit fractions. arXiv:2404.07113 (2024).

theorem Erdos47.erdos_47 : True ↔ ∀ (δ : ℝ), 0 < δ → ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, δ * Real.log ↑N < A.reciprocalSum → ∃ S ⊆ A, S.reciprocalSum = 1

If $\delta>0$ and $N$ is sufficiently large in terms of $\delta$, and $A\subseteq\{1,\ldots,N\}$ is such that $\sum_{a\in A}\frac{1}{a}>\delta \log N$ then must there exist $S\subseteq A$ such that $\sum_{n\in S}\frac{1}{n}=1$?

Bloom [Bl21] proved this in the affirmative.