Problem 339

../

claims/: The 1 claim page of Problem 339, one per claimant's result; the problem's standing derives from them.

---

Statement. Let $A\subseteq \mathbb{N}$ be a basis of order $r$. Must the set of integers representable as the sum of exactly $r$ distinct elements from $A$ have positive lower density?

Status. The site labels the problem proved (page last edited 2025-10-14): Hegyvári, Hennecart and Plagne [HHP03] answered it, and the companion upper-density question of Erdős and Graham, in the affirmative. The accepted claim is Hegyvári, Hennecart and Plagne 2003.

Source. erdosproblems.com/339, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #339, https://www.erdosproblems.com/339.

References.

• [HHP03] Hegyvári, N. and Hennecart, F. and Plagne, A., A proof of two Erdős' conjectures on restricted addition and further results. J. Reine Angew. Math. (2003), 199-220.

Formalization. None recorded on the site, and the community database records the problem unformalized. Boris Alexeev's lean-proofs repository holds a Lean 4 file that declares itself a formalization of Hegyvári, Hennecart and Plagne's solution, with the AI systems Codex and GPT-5.6 Sol as its formal authors, linked at its pinned commit from the claim page; this corpus has not built it.

Current assessment

Proved by a refereed paper of 2003. The site's formulation above (export of 2026-09-04, page read) asks whether, for a basis $A$ of order $r$, the integers that are sums of exactly $r$ distinct elements of $A$ have positive lower density. The answer is yes, proved by Hegyvári, Hennecart and Plagne in J. Reine Angew. Math. 560 (2003), 199--220, which also proves the companion statement that positive upper density of the $r$-fold sums implies positive upper density of the restricted sums; the claim page records the acceptance. The paper is not held here, its theorem numbering is not recorded, and its proof is not compiled or reviewed here.

Scope. Read: the site's problem page, the Crossref record of the paper, and the header and theorem statement of the Lean file. Not done: reading the paper, building the Lean file, and any literature search beyond these sources.