The existence of small prime gaps in subsets of the integers

Abstract

Abstract. We consider the problem of finding small prime gaps in various sets C ⊂ N. Following the work of Goldston-Pintz-Yıldırım, we will consider collections of natural numbers that are wellcontrolled in arithmetic progressions. Letting qn denote the n-th prime in C, we will establish that for any small constant ǫ > 0, the set {qn|q n+1 − qn ≤ ǫ log n} constitutes a positive proportion of all prime numbers. Using the techniques developed by Maynard and Tao we will also demonstrate that C has bounded prime gaps. Specific examples, such as the case where C is an arithmetic progression have already been studied and so the purpose of this paper is to present results for general classes of sets.

Narrow Progressions in the Primes

“…We set aside (29) for now and work on (30). Bounding all the components of f in magnitude by ν + 1 and using Proposition 3, we see that (1) and the claim follows.…”

“…We first dispose of the "globally bad" primes, in which p divides the entire polynomial Q j − Q j ′ . As Q j − Q j ′ is non-constant and has coefficients O(W O(1) ), we see that the product of all such primes is O(W O (1)…”

“…The , where the implied constants in the exponents depend on K. It is at this point that we crucially use the hypothesis that L be large compared with K, to ensure that this error is still o (1). Thus (26) can be written as ,d;k=1,...,K;ω=0,1…”

“…One could obtain precise asymptotics on the left-hand side using the calculations of Gallagher [7], but we can obtain a crude upper bound that suffices as follows. From (1) we see that In the converse direction, if we use the "Cramér random model" of approximating the primes P ∩ [N] by a random subset of [N] of density 1/ log N, we can asymptotically almost surely match this lower bound, thanks to the work of Conlon-Gowers [6] and Schacht [17]: Proposition 2. Let k ≥ 2 and δ , ε > 0, let C > 0 be sufficiently large depending on δ , k, and suppose that N is sufficiently large depending on k, δ , ε.…”

“…It is thus natural to conjecture that in Theorem 2 one can take r to be as small as log k−1+o(1) N. Remark 1. If one seeks progressions inside the full set P of primes, rather than of dense subsets of the primes, then the Hardy-Littlewood prime tuples conjecture [13] predicts that one can take r to be of size O k (1); indeed, one should be able to take r to be the product of all the primes less than or equal to k. In the case k = 2, the claim that one can take r = O(1) amounts to finding infinitely many bounded gaps between primes, a claim that was only recently established by Zhang [20]. For higher k, the claim r = O k (1) appears to currently be out of reach of known methods; the best known result in this direction, due to Maynard [15] (and also independently in unpublished work of the first author), shows that for any sufficiently large R > 1, there exist infinitely many intervals of natural numbers of length R that contain ≥ c log R primes for some absolute constant c > 0, but this is too sparse a set of primes to expect to find length k progressions for any k ≥ 3.…”