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Source. Proposition 12, p. 15 (Section 10.1, pp. 15--16), of Infinite Deletions from Strongly Minimal Additive Bases, manuscript (2026), no author printed, posted by Svyable in the thread of Erdős Problem 881 on 2026-05-03, https://www.overleaf.com/read/dckvqtggbjzn; the edition read is identified on the source card.

Read depth. Claims checked: the statement was read clause by clause on the print and the proof on pp. 15--16 was read; no step is checked here.

Statement

$\mathbb N=\{0,1,2,\ldots\}$ and $hA$ is the set of sums of exactly $h$ elements of $A$, repetitions allowed (p. 2).

Proposition 12 (p. 15). No set $A\subset\mathbb N$ is both an asymptotic basis of order $2$ and minimal as an asymptotic basis of order $3$.

The paper states this as the case $k=2$ of a "no-booster variant": whether some $A$ is a basis of order $k$ and minimal as a basis of order $k+1$. It says (p. 16) that for $k\ge3$ this question appears to be distinct from its construction, and does not settle it.

Proof pointer

Pp. 15--16. Minimality at order $3$ gives every $a\in A$ arbitrarily large witnesses in $3A\setminus3(A\setminus\{a\})$. Subtracting another element $b$ from such a witness and using the order-$2$ basis property, the proof argues that every representation of $a+b$ as a sum of two elements uses $a$, so $A$ is a Sidon set. Then $|A\cap[0,x]|\le(1+o(1))x^{1/2}$, so $2A\cap[0,x]$ has at most $(\frac12+o(1))x$ elements, which is incompatible with $A$ being a basis of order $2$.

Bears on

• Problem 881: not a result on the problem's question. The paper (Sections 10.1 and 10.3, pp. 15--16) sets it beside its construction: without the booster, the question whether a set is a basis of order $k$ and a minimal basis of order $k+1$ has answer no for $k=2$ by this proposition; the paper does not settle it for $k\ge3$.