---

Source. Proposition 4.1, p. 8, proved on pp. 8–9, 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 4.1 (p. 8). For every $C>0$ there is a set $E\subseteq\mathbb N$ such that $E$ is an additive basis of order 3, $R_{E,3}(n)\ge C\log n$ for all sufficiently large $n$, and $E$ contains no minimal additive basis of order 3.

Bases and $R_{E,3}$ are in the at-most-three sense of Theorem 1.1.

Proof pointer

Pp. 8–9. Choose $J$ with $J\eta_3>C$ and distinct positive integers $P=\{p_1,\ldots,p_J\}$, let $F=\{2p_j,2p_j+1:1\le j\le J\}$, apply Proposition 3.4 to the list of all pairs $(U,V)$ with $\varnothing\ne U\subseteq P$ and $V\subseteq P+P$ and with $P_0=P$, and put $E=2A\cup F$. Writing $q-p_j=a+b$ gives $2q+r=2a+2b+(2p_j+r)$, so $E$ is an order-3 basis with at least $J\eta_3\log q-O_P(1)$ representations of $2q+r$. For a subbasis $T$, the odd and even fillers in $T$ determine a pair $(U,V)$ for which $\Phi_{U,V}(D)$ is cofinite, $D=\{a\in A:2a\in T\}$; Proposition 3.4 (5) then gives $d\notin P$ with $T\setminus\{2d\}$ still an order-3 basis.

Dependencies

Proposition 3.4. Read depth: claims checked; the statement and proof were read clause by clause on the print.

Bears on

• Problem 870: the case $k=3$ of Theorem 1.1, with the problem site's at-most-$k$ count of representations.