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Claim. Let $A\subseteq\mathbb{N}$ be eventually periodic: for some period $m$, cutoff $N$ and pattern $S\subseteq G=\mathbb{Z}/m\mathbb{Z}$, an integer $n\ge N$ lies in $A$ exactly when its residue lies in $S$. Let $E=A\cap[1,N-1]$ be the exceptional elements, $H$ the subgroup of $G$ generated by $S$, and $\Sigma(E)$ the set of residues of the sums over subsets of $E$. The report's Theorem 2 states that $A$ is an asymptotic basis exactly when $S$ and the residues of $E$ together generate $G$; that $A$ has a restricted order exactly when $H+\Sigma(E)=G$; that $A\setminus F$ is a basis for every finite $F$ exactly when $H=G$, in which case the restricted order is at most $m$; and that if every $A\setminus F$ has the same order $h$, then the restricted order equals $h$. Its Theorem 1 states that $A=\{4,6\}\cup\{10q:q\ge1\}\cup\{10q+3:q\ge1\}$ is an asymptotic basis of order exactly $3$ with restricted order exactly $6$, and stays a basis after the removal of any finite set; the restricted order of a basis of order $3$ can therefore be at least $6$, where the construction of Hegyvári, Hennecart and Plagne [HHP07] gives $4$. The report, dated 2026-07-28 on the Erdős Problem a Day site, is marked partial and signed by Patrick White with Claude (Anthropic) named as the AI system used; it carries elementary proofs of both theorems. A forum comment of 2026-08-17 relayed it to the problem's thread.

Covers. The eventually periodic sets, for the first question of the statement of Problem 338, the condition for a restricted order to exist, and for the two questions of the site's remarks: whether a set that stays a basis after every finite removal has a restricted order, yes with the period as a bound, and whether the restricted order equals the order when every such removal leaves a basis of the same order, yes. For the third question of the statement the report gives only the sufficient condition of the last part. Nothing is claimed for sets that are not eventually periodic, and the second question of the statement, a bound on the restricted order in terms of the order, is not claimed: the order-3 example shows only that such a bound at order $3$ is at least $6$.

Standing. Claimed. The site's label is OPEN (page last edited 2025-09-14); as of 2026-10-06 no examination of the report by a named mathematician is recorded, and there is no refereed publication and no Lean development. Veljjanoski's write-up of 2026-10-01, recorded at Veljjanoski 2026, credits the report with the eventually periodic case and recovers its existence part for sets of positive lower density.

Depends on. No page of this wiki.