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Statement

The paper's definition (p. 1): "A $\mathcal{P}$-set is a set $\mathcal{S}$ of positive integers such that no element of $\mathcal{S}$ divides the sum of any two (not necessarily being different) larger elements." Write $A_S(N)=|\{s\in S:s\le N\}|$ (p. 1). From the end of the introduction (p. 2) the paper supposes that $S$ is a P-set of pairwise coprime integers. Theorem (p. 2). Let any $\varepsilon>0$ be given. Then

$$A_S(N)<(3+\varepsilon)N^{2/3}(\log N)^{-1}$$

for infinitely many integers $N$.

The introduction (p. 1) attributes to Schoen [6] the bound $A_S(N)<2N^{2/3}$ for infinitely many $N$ in the same pairwise coprime case, by the analytic large sieve, and the remark that "by giving a counterexample, Schoen pointed out that $c$ cannot be choosen [sic] greater than $1/2$" in the Erdős--Sárközy conjecture $A_S(N)<N^{1-c}$ infinitely often. Both are quoted here second-hand; Schoen's paper is not held.

Source. S. Baier, A note on P-sets, Integers 4 (2004), #A13, 6 pp. (received 3 October 2003, accepted 26 September 2004, published 8 October 2004, per the paper's header; listed on the journal's volume 4 page); the Theorem on p. 2 of the retained journal PDF, read in the text layer.

Read depth. Claims checked: the definition, the Theorem and the introduction's attributions were read clause by clause in the text layer. The proof (Sections 2--3, a sieve with the coprime elements of $S$ as moduli, Montgomery's arithmetic large sieve as Lemma 1, and mean values of multiplicative functions) was read for structure only.

Proof pointer

If $q<r$ are in $S$ then no $s\in S$ with $s>q$ lies in the class $-r$ modulo $q$, so each $q\in S$ excludes at least $1+[q/2]$ residue classes modulo $q$ from the elements of $S$ above $q$ (p. 2). With the elements of $S$ up to $z$ as moduli, the arithmetic large sieve bounds the number of elements of $S$ in $(z,N]$; a mean-value estimate for the resulting multiplicative function gives the theorem along a suitable sequence of $N$. Not reconstructed here.

Dependencies

Montgomery's arithmetic large sieve (the paper's Lemma 1), taken at statement level.

Bears on

• Problem 12: the best upper bound in hand for the pairwise coprime case, quoted by the site as "Baier has improved this to $\ll N^{2/3}/\log N$"; it says nothing about general sets with property P, for which the 2026 constructions give $N/(\log N)^{O(\log\log\log N)}$ for all large $N$.