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Source. Proposition 3.4, pp. 5–6, with the definition of $\Phi_{U,V}$ on p. 4 and Lemmas 3.1–3.3 (pp. 4–5), of David Turturean, A Negative Answer to Erdős Problem #870, preprint dated April 2026 (11 pp.), https://www.overleaf.com/read/gknkvvxrymfv; the edition read is named on the source card.

Setting

P. 4. For finite $U,V\subseteq\mathbb Z_{\ge0}$ and $D\subseteq\mathbb N$, $\Phi_{U,V}(D)=\bigl(U+(D\cup(D+D))\bigr)\cup(V+D)$, so $x$ lies in it when $x-u\in D\cup(D+D)$ for some $u\in U$ or $x-v\in D$ for some $v\in V$. $r_A$ and $A(x)$ are as in Proposition 2.3.

Statement

Proposition 3.4 (pp. 5–6). There is an absolute constant $\eta_3>0$ with the following property. Let $\mathcal P=\{(U_\lambda,V_\lambda):\lambda\in\Lambda\}$ be any finite list of pairs of finite subsets of $\mathbb Z_{\ge0}$, each $U_\lambda$ nonempty, and let $P_0\subset\mathbb N$ be finite. There is a set $A\subseteq\mathbb N$ such that

1. $A\cap P_0=\varnothing$;
2. $A+A$ is cofinite;
3. $r_A(n)\ge\eta_3\log n$ for all sufficiently large $n$;
4. $A(x)=o(x)$;
5. for every $(U,V)\in\mathcal P$, every $D\subseteq A$ and every finite exceptional set $F_0\subseteq D$: if $\Phi_{U,V}(D)$ is cofinite, then there is $d\in D\setminus F_0$ such that, with $D'=D\setminus\{d\}$, the set $\Phi_{U,V}(D')$ is still cofinite and $D'$ is an additive basis of order 3.

The proof (p. 7) takes $\eta_3=15/(512\log2)$.

Proof pointer

Pp. 6–8. The Larsen–Larsen construction is run with lag 10 and a Bernoulli constant chosen in terms of $\mathcal P$, but each canary is replaced by a cluster $y-u$, $u\in U_\lambda$, around a random center $y$, with restoration elements $y-u-s$ for old control summands $s$. Lemma 3.2 supplies, for each robust element, many representations whose summands avoid the fixed shift differences, which rules out same-cluster accidental representations; Lemma 3.1, a hypergeometric and difference-set estimate, gives a summable Borel–Cantelli bound that rules out accidental representations across clusters and keeps the points $y-v$, $v\in V$, out of $A$. Lemma 3.3 makes fixed differences between canaries outside $\Omega_B-\Omega_B$ occur only finitely often, which gives the order-3 property of $D'$. The set $P_0$ is avoided by deleting finitely many Bernoulli variables at the outset.

Dependencies

Lemmas 3.1–3.3 and the construction of D. Larsen and M. Larsen, Robust additive bases without minimal subbases, arXiv:2601.18507 (2026), whose Lemmas 2, 6 and 7, Proposition 5, p. 8 bound and finite-incidence argument the paper cites.

Read depth: claims checked. The statement was read clause by clause on the print and the proof followed in outline; the cited Larsen–Larsen results were not read.

Bears on

• Problem 870: the input of Proposition 4.1, which gives the case $k=3$ of Theorem 1.1. On its own it gives no order-3 basis without a minimal subbasis.