Problem 298

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claims/: The 1 claim page of Problem 298, one per claimant's result; the problem's standing derives from them.

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Statement. Does every set $A\subseteq \mathbb{N}$ of positive density contain some finite $S\subset A$ such that $\sum_{n\in S}\frac{1}{n}=1$?

Formulation. "Positive density" has three readings: the natural density of $A$ exists and is positive, the lower density of $A$ is positive, or the upper density $\limsup_{N\to\infty}|A\cap[1,N]|/N$ is positive. Each of the first two implies the third, so the upper-density reading is the strongest statement, and a yes under it answers every reading. No text of the poser fixes one reading. The 1980 monograph [ErGr80], printed p. 36, states the conjecture for every sequence of positive density. Erdős's 1992 paper [Er92c], printed p. 46, asks whether $f(n)/n\to0$, where $f(n)$ is the least length for which every sequence $1\le x_1<\cdots<x_{f(n)}\le n$ makes his equation (30), $\sum_i\varepsilon_i/x_i=1$ with $\varepsilon_i\in\{0,1\}$, solvable, and adds "In other words: Is it true that (30) is solvable in every sequence of positive lower density?". Its main question, $f(n)/n\to0$, implies the upper-density reading, so the paper's two sentences point to different readings. The site's commentary says that Bloom's proof covers positive upper density, which it calls likely to be what Erdős intended.

Status. Proved, in the site's label. Bloom's theorem applies under the assumption of positive upper density, so it answers the Statement in each of its readings. The site's label is PROVED (LEAN); the existing formal proof and statement-only declarations are distinguished below.

Source. T. F. Bloom, Erdős Problem #298, https://www.erdosproblems.com/298, accessed 2026-09-05. The site's original references are [ErGr80] and [Er92c]. As of that date the discussion and proof-claim pages had no comments or proof claims.

References.

• [ErGr80] P. Erdős and R. L. Graham, Old and new problems and results in combinatorial number theory, Monographies de L'Enseignement Mathématique (1980). Bloom's paper locates the density question on p. 36.
• [Er92c] P. Erdős, Some of my forgotten problems in number theory, Hardy–Ramanujan Journal 15 (1992), 34–50. Original reference listed by the site.
• [Bl21] T. F. Bloom, On a density conjecture about unit fractions, arXiv:2112.03726 (2021), v2 (2023), with Appendix B co-written by T. F. Bloom and B. Mehta; JEMS 27 (2025), 4563–4589.
• [LiSa24] Y. P. Liu and M. Sawhney, On further questions regarding unit fractions, arXiv:2404.07113v1 (10 April 2024), Theorem 1.1, p. 1; proof pp. 19–20.

Formalization. The Bloom–Mehta Lean 3 proof, its Lean 4 ports and the statement-only declarations are described under Existing formalization; the corpus has built none of them.

Current assessment

Claims. One accepted full claim settles the problem: Bloom's Theorem 2, whose acceptance evidence is the refereed publication in J. Eur. Math. Soc. 27 (2025). The site's curator is the theorem's author, so the site's label is recorded on the claim page as the catalog's label and not as independent review; the author's own Lean 3 formalization is a link on that page and, unbuilt here, gives no formalized evidence. The frontmatter standing is derived from the claim page. Liu and Sawhney's sharper threshold below refines the finite criterion; that threshold is the subject of Problem 47, where Bloom's Theorem 3 and Liu and Sawhney's Theorem 1.1 are the accepted claim pages Bloom's reciprocal-mass threshold and Liu and Sawhney's four-fifths threshold.

Even a positive upper density suffices; existence of a natural density is unnecessary. The proof and its essential dependencies are compiled on the result pages of the source card. The proof uses the explicitly identified variant in the existing formalization to handle a parameter mismatch in the printed technical proposition; see the source discrepancy. This compilation detail does not change the solved mathematical status.

The Liu–Sawhney threshold below is the strongest threshold known to the corpus. The linked full proof uses arXiv:2404.07113v1 (10 April 2024), with explicit sufficient replacements for the false printed multiplicity-counting claim in Lemma 2.2 and the false unrestricted form of Lemma 5.1. It does not certify those literal statements. The later published PDF has not been compared; these are compilation corrections, not attributed author errata. Bloom's earlier solution of the qualitative problem remains independent of this refinement.

No independent review of either route is recorded.

Progress and known results

Bloom's 2021 solution, Theorem 2 of arXiv v2 (Theorem 1.2 of the published version), proves

$$\limsup_{N\to\infty}\frac{|A\cap[1,N]|}{N}>0 \quad\Longrightarrow\quad \exists S\subseteq A\text{ finite}:\ \sum_{n\in S}\frac1n=1.$$

Theorem 3 (Theorem 1.3 of the published version) gives a finite quantitative criterion: for an absolute $C$ and sufficiently large $N$, a set $A\subseteq\{1,\ldots,N\}$ of reciprocal mass at least $C\log N\log\log\log N/\log\log N$ has a unit subsum. The source digest explains the refinement of Croot's Fourier and smooth-number method and records Pomerance's complementary obstruction construction.

Liu and Sawhney's later Theorem 1.1 improves the quantitative threshold: for every $\varepsilon>0$, there is $N_0(\varepsilon)$ such that, for every integer $N\ge N_0(\varepsilon)$ and every $A\subseteq\{1,\ldots,N\}$,

$$\sum_{n\in A}\frac1n\ge(\log N)^{4/5+\varepsilon} \quad\Longrightarrow\quad \exists S\subseteq A:\ \sum_{n\in S}\frac1n=1.$$

The site links the related Problem 46 and Problem 47. The bounded-gap consequence is recorded separately for Problem 299.

Existing formalization

The Google DeepMind file contains upper-density and natural-density statements, whose proof bodies are sorry. Its external proof tag links the Bloom–Mehta Lean 3 project. The accessible solution is unit_fractions_upper_density. Appendix B of Bloom's paper reports complete formal verification. Two Lean 4 postings of the same formalization are linked on the claim page: the file src/latest/ErdosProblems/Erdos298.lean of Boris Alexeev's lean-proofs collection, which names Bloom as informal author and Mehta and Bloom as formal authors and proves erdos_298 and erdos_298_density from unit_fractions_upper_density, and a vendored single-file copy of the Lean 4 port in Jayyhk/erdos-lean. The corpus has built none of these. No formalization of the stronger Liu–Sawhney threshold was located.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

• erdos_1997_some_my_favorite_problems_results / display_4_4
• bloom_2021_density_conjecture_about_unit_fractions
• bloom_2021_density_conjecture_about_unit_fractions / bounded_gaps
• bloom_2021_density_conjecture_about_unit_fractions / corollary_1
• bloom_2021_density_conjecture_about_unit_fractions / lemma_1
• bloom_2021_density_conjecture_about_unit_fractions / lemma_2
• bloom_2021_density_conjecture_about_unit_fractions / lemma_3
• bloom_2021_density_conjecture_about_unit_fractions / lemma_4
• bloom_2021_density_conjecture_about_unit_fractions / lemma_5
• bloom_2021_density_conjecture_about_unit_fractions / lemma_6
• bloom_2021_density_conjecture_about_unit_fractions / lemma_7
• bloom_2021_density_conjecture_about_unit_fractions / proposition_1
• bloom_2021_density_conjecture_about_unit_fractions / proposition_2
• bloom_2021_density_conjecture_about_unit_fractions / proposition_3
• bloom_2021_density_conjecture_about_unit_fractions / theorem_2
• bloom_2021_density_conjecture_about_unit_fractions / theorem_3
• bloom_2021_density_conjecture_about_unit_fractions / theorem_4
• liu_2024_further_questions_regarding_unit_fractions
• liu_2024_further_questions_regarding_unit_fractions / fact_2_5
• liu_2024_further_questions_regarding_unit_fractions / lemma_2_2
• liu_2024_further_questions_regarding_unit_fractions / lemma_2_3
• liu_2024_further_questions_regarding_unit_fractions / lemma_2_4
• liu_2024_further_questions_regarding_unit_fractions / lemma_2_6
• liu_2024_further_questions_regarding_unit_fractions / lemma_3_1
• liu_2024_further_questions_regarding_unit_fractions / lemma_5_1
• liu_2024_further_questions_regarding_unit_fractions / lemma_6_1
• liu_2024_further_questions_regarding_unit_fractions / lemma_6_2
• liu_2024_further_questions_regarding_unit_fractions / proposition_5_2
• liu_2024_further_questions_regarding_unit_fractions / theorem_1_1
• liu_2024_further_questions_regarding_unit_fractions / theorem_2_1