Bounded gaps between primes in number fields and function fields

The existence of small prime gaps in subsets of the integers

“…These results, which will be stated in section 6, differ from our own in the sense that we obtain small prime gaps frequently, as opposed to "infinitely often". For other results concerning small gaps in special sets of primes we refer the reader to [1], [16], [19].…”

Small gaps between configurations of prime polynomials

“…We need only consider d j built out of the prime divisors of k j=1 (f + h j ), of which there are at most 1/ǫ. Then there are at most 2 1/ǫ such choices for each d j , so our upper bound for (2) is at most Cn|A| for some new C > 0, establishing (1).…”

“…Although we will eventually need to fix w in the proof of Theorem 2, it is helpful to think of w tending to infinity up until that point, and we use the asymptotic notation o(1) for a quantity tending to zero as w → ∞. Since H is admissible, we can take w > deg(h k ) and fix a congruence class b (mod W ) with (W, b + h j ) = 1 for each h j ∈ H. We allow n → ∞, and we always insist that w ≪ log log n so that deg(W ) ≪ log n and terms that tend to zero as n → ∞ are also o (1).…”

“…These differ from the integer weights of Maynard in [6], but produce essentially the same results. A version of Proposition 1 for Maynard's weights in F q [t] appeared in [1], but it is unclear how to adapt their setup to obtain a concentration estimate comparable to Proposition 2. Such an estimate was the heart of Pintz' argument in [7] (see also [9,Proposition 14]).…”