Baier 2004 6

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theorem: Baier's sharpening of Schoen's large-sieve bound for P-sets of pairwise coprime integers.

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Stephan Baier, A note on P-sets. Integers 4 (2004), #A13, 6 pp.

A P-set is a set S of positive integers in which no element divides the sum of any two larger elements; Erdos and Sarkozy conjectured A_S(N) < N^{1-c} infinitely often. For P-sets of pairwise coprime integers Schoen had proved A_S(N) < 2N^{2/3} infinitely often via the analytic large sieve. The paper's single Theorem sharpens this to A_S(N) < (3+eps)N^{2/3}(log N)^{-1} for infinitely many N. The method sets up a sieve in which the coprime elements q of S play the role of primes, each excluding at least 1+[q/2] residue classes, and applies Montgomery's arithmetic form of the large sieve (Lemma 1) together with mean-value estimates for multiplicative functions. This bears directly on Erdos problem 12, the Erdos-Sarkozy question on how dense a P-set can be, giving the best known bound in the pairwise coprime case; Schoen's counterexample shows c cannot exceed 1/2.

The retained folder-name PDF is the journal's six-page file (Integers 4 (2004), paper A13; received 3 October 2003, accepted 26 September 2004, published 8 October 2004 per its header; listed on the journal's volume 4 page). Read status: claims checked for the P-set definition, the Theorem (p. 2) and the introduction's attributions to Schoen, read in the text layer; the sieve argument of Sections 2--3 was read for structure only. Result page: theorem. The paper's definition admits two equal larger elements ("not necessarily being different"). The journal's file prints no license line; the journal's site states "All works of this journal are licensed under a Creative Commons Attribution 4.0 International License" (https://math.colgate.edu/~integers/, read 2026-10-02): the Creative Commons Attribution 4.0 license, by the journal's current site-wide statement.

Source: https://math.colgate.edu/~integers/vol4.html.

Bears on. #12

Results to transcribe.

• Theorem: For every eps>0, any P-set S of pairwise coprime integers satisfies A_S(N) < (3+eps)N^{2/3}(log N)^{-1} for infinitely many integers N (p. 2; result page theorem).
• Lemma 1: Montgomery's arithmetic large sieve applied with a set of pairwise coprime moduli: |M| <= (N+Q^2)(sum_{k<=Q} g(k))^{-1}.
• Setup observation: If q,r in S with qq lies in the class -r mod q, so each q in S excludes at least 1+[q/2] residue classes mod q.