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Source. Proposition 2.3, p. 3, with Lemmas 2.1 (p. 2) and 2.2 (p. 3), 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

Pp. 1–3. $r_A(n)$ is the number of pairs $(a,b)\in A^2$ with $a\le b$ and $a+b=n$, the exact two-summand representations. An additive basis of order 2 in the at-most-two sense is a set $D$ with $D\cup(D+D)$ cofinite. $A(x)$ is the number of elements of $A$ up to $x$.

Statement

Proposition 2.3 (p. 3). There are an absolute constant $\eta_2>0$ and a set $A\subseteq\mathbb N$ such that

1. $A\cup(A+A)$ is cofinite;
2. $r_A(n)\ge\eta_2\log n$ for all sufficiently large $n$;
3. $A(x)=o(x)$;
4. $A$ contains no minimal additive basis of order 2 in the at-most-two sense.

Lemma 2.1 (p. 2). Almost surely the final set $A$ of the Larsen–Larsen construction satisfies $A(x)=o(x)$, uniformly in $x$ and not only at the stage boundaries $x=X_n=2^{2^n}$.

Lemma 2.2 (p. 3). Almost surely, for all sufficiently large $n$, there are no two distinct stage-$n$ canaries $b,b'\in B_n$, element $a$ of the earlier stages $A''(n-1)=\bigcup_{j<n}A''_j$ and element $s$ of $S_n$, the set holding the control summands of the stage-$n$ canaries, with $b'-b=a-s$.

Proof pointer

Pp. 2–4. The set is the random order-2 basis of Larsen and Larsen, built in stages $I_n=[X_n,X_{n+1})\cap\mathbb N$ by Bernoulli sampling, deletion of the summands of a sparse set $B_n$ of canaries, and addition of restoration elements. The exact order-2 basis property, the logarithmic lower bound and the absence of a minimal subbasis in the exact convention are cited from Larsen and Larsen. Lemma 2.1 gives (3), counting the Bernoulli samples by Chernoff bounds and the restoration elements by the doubly exponential growth of $X_n$. Lemma 2.2, a summable Borel–Cantelli bound, excludes a representation of a canary by another canary's restoration element and an old element, so each large canary keeps only its intended representations. The passage to the at-most-two convention shows that large canaries lie outside $A$, that every large robust element has at least two representations in any at-most-two subbasis $D$, and that deleting any $d\in D$ leaves an at-most-two basis.

Dependencies

Lemmas 2.1 and 2.2, and the construction of D. Larsen and M. Larsen, Robust additive bases without minimal subbases, arXiv:2601.18507 (2026), including its finite-incidence argument, which the paper cites. The constant $\eta_2$ is not given explicitly.

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

Bears on

• Problem 870: the order-2 input of Proposition 5.2, which gives the cases $k\ge4$ of Theorem 1.1. On its own it concerns order 2, outside the problem's range $k\ge3$.