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Source. Proposition 5.2, p. 9, proved on pp. 9–10, 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.

Statement

Proposition 5.2 (p. 9). For every integer $k\ge4$ and every $C>0$ there is a set $E\subseteq\mathbb N$ such that $E$ is an additive basis of order $k$, $R_{E,k}(n)\ge C\log n$ for all sufficiently large $n$, and $E$ contains no minimal additive basis of order $k$.

Bases and $R_{E,k}$ are in the at-most-$k$ sense of Theorem 1.1.

Proof pointer

Pp. 9–10. With $h=k-2$, take $A$ and $\eta_2$ from Proposition 2.3, $L$ with $L\eta_2>C$, and $N,M,F,\tau$ from Lemma 5.1, and put $E=MA\cup F$. An $h$-tuple of fillers fixes the residue and at most two elements of $A$ the quotient, which gives the basis property and, summed over $L$ tuples, the logarithmic count. For an order-$k$ subbasis $T$, the rigid residue $\tau$ forces $D=\{a\in A:Ma\in T\}$ to be an at-most-two basis; by Proposition 2.3 some $d\in D$ can be dropped, and $A(x)=o(x)$ shows that the fillers in $T$ reach every residue, so $T\setminus\{Md\}$ is still an order-$k$ basis.

Dependencies

Proposition 2.3 and Lemma 5.1. Read depth: claims checked; the statement and proof were read clause by clause on the print.

Bears on

• Problem 870: the cases $k\ge4$ of Theorem 1.1, with the problem site's at-most-$k$ count of representations.