Merikoski 2020 limit points normalized prime gaps

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corollary_2: Merikoski's measure bound: for every T > 0 the Lebesgue measure of the set of limit points of (p_{n+1} - p_n)/log p_n in [0,T] is at least T/3.

corollary_3: Merikoski's syndeticity result: there is a constant C >= 0 such that every interval [T, T+C] with T >= 0 contains a limit point of (p_{n+1} - p_n)/log p_n.

theorem_1: Merikoski's four-point theorem: for any reals beta_1 <= beta_2 <= beta_3 <= beta_4, some difference beta_j - beta_i with i < j is a limit point of (p_{n+1} - p_n)/log p_n.

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Merikoski, Jori, Limit points of normalized prime gaps. J. Lond. Math. Soc. (2) 102 (2020), 99-124. doi:10.1112/jlms.12314. The copy read for this card is the arXiv version (arXiv:1811.03008v3), whose record names arXiv's non-exclusive distribution license, every other right reserved.

Let L be the set of limit points of (p_{n+1} - p_n)/log p_n. Theorem 1 shows that for any reals beta_1 <= beta_2 <= beta_3 <= beta_4, the set L meets {beta_j - beta_i : 1 <= i < j <= 4}, improving the analogous statements for nine reals (Banks, Freiberg and Maynard) and five reals (Pintz). Corollary 2 deduces that the Lebesgue measure of L intersect [0,T] is at least T/3 for all T > 0, improving Pintz's (1/4 - o(1))T, and Corollary 3 gives a constant C with L meeting [T, T+C] for all T >= 0, so L is syndetic. The improvement comes from using Chen's sieve rather than Selberg's sieve to obtain a better upper bound for a certain sum over prime pairs, combined with a modified Bombieri-Vinogradov theorem (Section 2) and a modified Maynard-Tao sieve (Section 4). Section 6 (pp. 26--31), present in the arXiv version, corrects a mistake in the proofs of Lemmas 15 and 16; the author says the text before it agrees with the published article. This bears on problem 5, which asks whether every C >= 0 is a limit point of (p_{n+1} - p_n)/log n, the same set L since log p_n ~ log n; the paper states it as the Erdos conjecture that L = [0, infinity]. The paper does not resolve it but pushes the known measure of L up to at least one third and shows that L has no arbitrarily long gaps, while noting that besides 0 and infinity no individual real is known to lie in L.

Source: https://arxiv.org/abs/1811.03008.

Read status: claims checked for the results linked below, statements read clause by clause on the printed pages of arXiv v3 (Theorem 1 and Corollaries 2 and 3 on p. 2, Propositions 4 and 5 on pp. 3--4); no proof is checked step by step.

Bears on.

• #5: Theorem 1 shows that L meets the difference set of any four reals, Corollary 2 that L has measure at least T/3 in every [0,T], and Corollary 3 that for some fixed ineffective C, L meets [T, T+C] for every T >= 0. None of them places any given C > 0 in L or shows that L = [0, infinity]. Remark 2 (p. 6) says that the argument would give L = [0, infinity] if the paper's prime-pair bound (1.6) held with a constant below 2 in place of 3.99, and that by the parity principle this should be as hard as a lower bound for such a sum over prime pairs, which would itself imply L = [0, infinity].

Results.

• Theorem 1 (p. 2): For any reals beta_1 <= beta_2 <= beta_3 <= beta_4, the set L of limit points of normalized prime gaps intersects {beta_j - beta_i : 1 <= i < j <= 4}.
• Corollary 2 (p. 2): For all T > 0, the Lebesgue measure of L intersect [0,T] is at least T/3.
• Corollary 3 (p. 2): There is a constant C >= 0 with L intersect [T, T+C] nonempty for all T >= 0, so L is syndetic.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.