Svyable 2026 infinite deletions strongly minimal additive bases

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lemma_8: The manuscript's lemma that for k >= 2, a finite S in {2,3,4,...}, c in S and M >= 1, every large U admits a finite D and an integer p, both in (M,U), with p a sum of k elements of S, D and 1 and a sum of k+1 elements of S and D, but not a sum of k+1 elements of S, D and 1 that avoids c.

lemma_9: The manuscript's lemma that for k >= 2, a finite S in {2,3,4,...} and M >= 1, every large U admits a finite D and an integer q, both in (M,U), with q a sum of k elements of S, D and 1 but not a sum of k elements of S and D.

main_theorem_6: The manuscript's claimed answer to its Problem 5: yes for order one, and for every k >= 2 a set that is an asymptotic basis of order k, minimal at order k both for single and for infinite deletions, with no infinite deletion an asymptotic basis of order k+1, and with at most O_k(x^(1/k)) elements up to x.

proposition_12: The manuscript's proposition that no set of nonnegative integers is both an asymptotic basis of order 2 and a minimal asymptotic basis of order 3.

proposition_7: The manuscript's case k = 1 of its deletion question: every asymptotic basis of order 1 has an infinite subset whose removal leaves an asymptotic basis of order 2.

theorem_11: The manuscript's finite-booster normal form: if A is an asymptotic basis of order k >= 1 and no infinite deletion from A is an asymptotic basis of order k+1, then removing some finite F from A leaves a minimal asymptotic basis of order k+1.

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Infinite Deletions from Strongly Minimal Additive Bases, manuscript (2026), 17 pp., no author printed; posted by Svyable in the thread of Erdős Problem 881 on erdosproblems.com on 2026-05-03. No notice is printed in the file; the hosting service's terms (https://www.overleaf.com/legal, read 2026-10-02) say "We don't claim any ownership of your stuff" and grant readers of a shared project no license; the term is unstated. The manuscript is not refereed.

The manuscript poses Erdős Problem 881 as its Problem 5 (p. 3), with a basis of order $k$ meaning that every large integer is a sum of exactly $k$ elements of $A$ and with $0\in\mathbb N$ (p. 2): if $A$ is an asymptotic basis of order $k$ that is strongly minimal under infinite deletions at order $k$, must some infinite $B\subset A$ leave $A\setminus B$ an asymptotic basis of order $k+1$? Main Theorem 6 (p. 3) states the answer yes for $k=1$ (Proposition 7, p. 4) and no for every $k\ge2$, through a set $A=C\cup\{1\}$ that is a basis of order $k$, ordinarily minimal and strongly minimal at order $k$, with no infinite deletion a basis of order $k+1$, and with $|A\cap[0,x]|=O_k(x^{1/k})$. The construction (Section 7, pp. 9--11) gives each element of $C$ witnesses private at order $k+1$ (Lemma 8, p. 5), gives the element $1$ witnesses at order $k$ (Lemma 9, p. 7), and covers successive intervals by thin digit blocks (Lemma 10, p. 8); Section 8 (pp. 12--14) argues that it works. Theorem 11 (p. 14) states a normal form: every set answering the question no is a minimal basis of order $k+1$ plus a finite set. Proposition 12 (p. 15) states that no basis of order $2$ is a minimal basis of order $3$.

The site's curator wrote in the thread that the manuscript appears to have been generated by GPT, and a reader's AI check posted there reports gaps, among them that the "Moreover" clauses of Lemmas 8 and 9 are false as stated and that Theorem 11 is not proved; the claim page records them.

Read status: claims checked. The statements below were read clause by clause on the print; the proofs were read for their structure only and no step is checked here.

Source: https://www.overleaf.com/read/dckvqtggbjzn.

Bears on.

• #881: the manuscript's Problem 5 is the problem's question with a basis of order $k$ read as sums of exactly $k$ elements (the site's definitions allow sums of at most $k$ elements); Main Theorem 6 states the answer yes for $k=1$ and no for each $k\ge2$ in that reading.

Results.

• Main Theorem 6 (p. 3): yes for $k=1$; for every $k\ge2$ a set $A$ that is an asymptotic basis of order $k$, ordinarily minimal and strongly minimal under infinite deletions at order $k$, with $A\setminus B$ not a basis of order $k+1$ for every infinite $B\subset A$, and with $A(x)=O_k(x^{1/k})$.
• Proposition 7 (p. 4): every asymptotic basis of order $1$ has an infinite $B$ with $A\setminus B$ an asymptotic basis of order $2$.
• Lemma 8 (p. 5): for $k\ge2$, finite $S\subset\{2,3,4,\ldots\}$, $c\in S$ and $M\ge1$, every large $U$ gives a finite $D\subset(M,U)$ and $p\in(M,U)$ with $S'=S\cup D$, $p\in k(S'\cup\{1\})\cap(k+1)S'$ and $p\notin(k+1)\bigl((S'\setminus\{c\})\cup\{1\}\bigr)$, with a "Moreover" clause placing $p$ and $D$ in any prescribed subinterval of $(M,U)$ whose length tends to infinity with $U$.
• Lemma 9 (p. 7): for $k\ge2$, finite $S\subset\{2,3,4,\ldots\}$ and $M\ge1$, every large $U$ gives a finite $D\subset(M,U)$ and $q\in(M,U)$ with $q\in k(S'\cup\{1\})$ but $q\notin kS'$, $S'=S\cup D$, with the same "Moreover" clause.
• Lemma 10 (p. 8), not given a page: for every $L\ge1$ a finite $Q_L\subset[0,O_k(L)]$ with $|Q_L|=O_k(L^{1/k})$ and $[0,L]\subset kQ_L$; and, for $T>kM$, a finite $D\subset(M,O_k(T+L))$ with $[T,T+L]\subset kD$ and $|D|=O_k(L^{1/k})$.
• Theorem 11 (p. 14): for $k\ge1$, if $A$ is an asymptotic basis of order $k$ and no infinite deletion from $A$ is an asymptotic basis of order $k+1$, then $A=C\cup F$ with $F\subset A$ finite and $C=A\setminus F$ a minimal asymptotic basis of order $k+1$.
• Proposition 12 (p. 15): no $A\subset\mathbb N$ is both an asymptotic basis of order $2$ and minimal as an asymptotic basis of order $3$.

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