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Source. Lemma 8, p. 5 (Section 5.1, pp. 5--7), 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. 5--7 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 8 (p. 5). Let $k\ge2$, let $S\subset\{2,3,4,\ldots\}$ be finite, $c\in S$, and $M$ a positive integer. For all sufficiently large $U$ there are a finite set $D\subset(M,U)$ and an integer $p\in(M,U)$ such that, with $S'=S\cup D$,

$$p\in k(S'\cup\{1\}),\qquad p\in(k+1)S',\qquad p\notin(k+1)\bigl((S'\setminus\{c\})\cup\{1\}\bigr).$$

The lemma continues (p. 5, quoted): "Moreover, once $U$ is large enough, $p$ 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. 5--7. $D$ consists of new elements $x_1,\ldots,x_k,y_1,\ldots,y_{k-1}$ with $p=c+x_1+\cdots+x_k=1+y_1+\cdots+y_{k-1}$, the $x_i$ and $y_1,\ldots,y_{k-2}$ free and $y_{k-1}$ determined by them. A representation of $p$ by $k+1$ terms avoiding $c$ is an affine equation in the free variables; comparing coefficients, the proof argues that none of these equations is an identity, so free variables chosen in a large box off finitely many hyperplanes give the third property. For the "Moreover" clause the proof offers one sentence (p. 5): the box may be shifted and rescaled into any large subinterval. On the same page it notes that variables near a parameter $X$ give $y_{k-1}=2X+O(1)$ and $p=kX+O(1)$.

A reader's AI check posted in the problem's thread, recorded on the claim page, reports that the "Moreover" clause 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 any element of $C$ destroys the basis property at order $k+1$.