Problem 336

For $r\geq 2$ let $h(r)$ be the maximal finite $k$ such that there exists a basis $A\subseteq \mathbb{N}$ of order $r$ (so every large integer is the sum of at most $r$ integers from $A$ ) and exact order $k$ (so every large integer is the sum of exactly $k$ integers from $A$ ).

Find the value of

\lim_r \frac{h(r)}{r^2}.

Source: erdosproblems.com/336

Number theory
Additive bases

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ClaimedAnswered

A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.

Claimed; the site's label is OPEN (page last edited 2025-10-28). One pending full claim is recorded: Snyder's Lean proof that the limit is one third, a Lean 4 development posted on 2026-07-15 and entered the same day on the site's proof-claims thread, stating that the maximal exact order over bases of order at most $r$ is attained and that $h(r)/r^2\to1/3$ ; the author reports the three standard axioms, the thread showed no comments on it as of 2026-10-06, and the development is not built or audited in this corpus. The published bounds are $1/3\le\liminf h(r)/r^2$ (Grekos [Gr88]) and $\limsup h(r)/r^2\le1/2$ (Nash [Na93]).

Problem 336

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