Banks 2016 limit points sequence normalized prime gaps

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corollary_1_2: The set L of limit points of the normalized prime gaps (p_{n+1} - p_n)/log p_n meets [0,T] in Lebesgue measure at least (1 - o(1))T/8 as T tends to infinity, with an ineffective o(1), and in measure greater than T/22 for every T > 0.

theorem_1_1: Banks, Freiberg and Maynard's main theorem: for any nine nonnegative reals beta_1 <= ... <= beta_9, at least one difference beta_j - beta_i with i < j is a limit point of the normalized prime gaps (p_{n+1} - p_n)/log p_n.

theorem_1_3: For each fixed integer m >= 2 and any 8m^2 + 8m nonnegative reals beta_1 <= ... <= beta_{8m^2+8m}, some vector of differences along an increasing chain of m + 1 indices is a limit point of the vectors of m consecutive normalized prime gaps.

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Banks, William D. and Freiberg, Tristan and Maynard, James, On limit points of the sequence of normalized prime gaps. Proc. Lond. Math. Soc. (3) 113 (2016), 515-539, doi:10.1112/plms/pdw036. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1404.5094), every other right reserved. The copy read for this card is arXiv:1404.5094v2 (20 October 2014), 25 pages; labels and pages on this card and its result pages are that version's.

Let L be the set of limit points of (p_(n+1) - p_n)/log p_n; the paper states Erdos's conjecture as L = [0, infinity] (p. 1). Before this work 0 and infinity were the only points known to lie in L (p. 1), although Erdos and Ricci had shown that L has positive Lebesgue measure, Hildebrand and Maier that lambda([0,T] cap L) >= cT for all sufficiently large T, and Pintz that L contains [0,c] for an ineffective c > 0 (p. 2). Theorem 1.1 (p. 2) proves that for k = 9 and any nonnegative reals beta_1 <= ... <= beta_9, at least one of the differences beta_j - beta_i (1 <= i < j <= 9) lies in L. Corollary 1.2 (p. 2) deduces lambda([0,T] cap L) >= (1 - o(1))T/8 as T -> infinity, with an ineffective o(1), which the paper reads as at least 12.5% of nonnegative reals lying in L, and the effective bound lambda([0,T] cap L) > T/22 for all T > 0. Theorem 1.3 (p. 3) is the version for chains of gaps: for each fixed integer m >= 2 and any 8m^2 + 8m nonnegative reals beta_1 <= ... <= beta_(8m^2+8m), some vector (beta_J(2) - beta_J(1), ..., beta_J(m+1) - beta_J(m)) with J(1) < ... < J(m+1) is a limit point in [0, infinity]^m of the vectors of m consecutive normalized gaps; the paper calls Theorem 1.1 a stronger version of its case m = 1. The method combines the Erdos-Rankin construction of long runs of composites (Section 5, pp. 17-22) with a uniform version of the Maynard-Tao theorem (Section 4, pp. 6-17), whose Theorem 4.3 (p. 10) rests on a modified Bombieri-Vinogradov theorem, Theorem 4.2 (p. 8); Section 6 (pp. 22-24) deduces Theorems 1.3 and 1.1. Section 7 (p. 24) remarks that if Theorem 4.2 held with an arbitrary fixed theta in (0,1), a minor adaptation of the Maynard-Tao argument would allow k = 5 in Theorem 1.1.

Source: https://arxiv.org/abs/1404.5094.

Read status. Claims checked: Theorem 1.1, Corollary 1.2 and Theorem 1.3 (pp. 2-3) were read clause by clause on the printed pages, and the proof of Corollary 1.2 (pp. 2-3) was read through. The deductions of Section 6 (pp. 22-24) were read for structure only, and Theorem 4.3 (p. 10) and Lemma 5.2 (p. 19) as statements; no other proof was checked, and nothing here is independently reviewed.

Bears on.

• #5: the problem asks, for each C >= 0, for a sequence along which (p_(n+1) - p_n)/log n tends to C; since log p_n/log n -> 1, the C for which such a sequence exists are exactly the finite points of L (an observation of the result pages, not of the paper). Theorem 1.1 puts some difference of any nine nonnegative reals among them, and Corollary 1.2 bounds their measure in [0,T] from below; neither names a particular C, so neither settles an instance of the problem. Theorem 1.3 concerns chains of m >= 2 gaps and is context only.

Results.

• Theorem 1.1 (p. 2): among any nine nonnegative reals some difference lies in L.
• Corollary 1.2 (p. 2): lambda([0,T] cap L) >= (1 - o(1))T/8 as T -> infinity, and lambda([0,T] cap L) > T/22 for all T > 0.
• Theorem 1.3 (p. 3): the version for chains of m >= 2 consecutive gaps, from 8m^2 + 8m nonnegative reals.

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