---

Source. Lemma 9, p. 7 (Section 5.2, pp. 7--8), 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. 7--8 was read; no step is checked here.

Statement

$hX$ is the set of sums of exactly $h$ elements of $X$, repetitions allowed (p. 2).

Lemma 9 (p. 7). Let $k\ge2$, let $S\subset\{2,3,4,\ldots\}$ be finite and $M$ a positive integer. For all sufficiently large $U$ there are a finite set $D\subset(M,U)$ and an integer $q\in(M,U)$ such that, with $S'=S\cup D$,

$$q\in k(S'\cup\{1\})\qquad\text{but}\qquad q\notin kS'.$$

The lemma continues (p. 7, quoted): "Moreover, once $U$ is large enough, $q$ and all elements of $D$ may be chosen inside any prescribed subinterval of $(M,U)$ of length tending to infinity with $U$."

Proof pointer

Pp. 7--8. Take new elements $u_1,\ldots,u_{k-1}>M$, $D=\{u_1,\ldots,u_{k-1}\}$ and $q=1+u_1+\cdots+u_{k-1}$. A representation of $q$ by $k$ terms of $S'$ is an affine equation in the $u_i$; matching coefficients would leave one term of $S$ equal to $1$, so none is an identity, and distinct $u_i$ off finitely many hyperplanes in the prescribed subinterval give $q\notin kS'$.

A reader's AI check posted in the problem's thread, recorded on the claim page, reports that this lemma's analogue of the "Moreover" clause of Lemma 8 is false as stated. The construction applies it on p. 10.

Bears on

• Problem 881: an ingredient of the manuscript's construction for Main Theorem 6; its witnesses are meant to show that deleting the element $1$ destroys the basis property at order $k$.