The Bombieri-Vinogradov theorem for nilsequences

Abstract

We establish results of Bombieri-Vinogradov type for the von Mangoldt function Λ(n) twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase e(P (n)); the results obtained are as strong as the ones previously known in the case of linear exponential twists. We derive a number of applications of these results. Firstly, we show that the primes p obeying a "nil-Bohr set" condition, such as αp k < ε, exhibit bounded gaps. Secondly, we show that the Chen primes are well-distributed in nil-Bohr sets, generalizing a result of Matomäki. Thirdly, we generalize the Green-Tao result on linear equations in the primes to primes belonging to an arithmetic progression to large modulus q ≤ x θ , for almost all q.

A transference principle for systems of linear equations, and applications to almost twin primes

“…Proofs of Theorem 7.1 and 7.2. We deduce Theorem 7.1 from [28,Theorem 4.3]; the deduction of Theorem 7.2 from [28, Theorem 4.4] is completely similar. Write ψ(n) = ϕ(g(n)).…”

“…It now follows (from [28,Lemma 3.6]) that {g(n)} n≤x is not totally η −O s,∆ (c) -equidistributed, which is a contradiction if c is chosen small enough. Let ψ (n) = ϕ(g (n)).…”

“…We will loosely call such nilsequences η-equidistributed. We caution that this notation is slightly different from [28], in that we do not require G/Γ F = 0 (where the integral is taken with respect to the unique Haar measure on G/Γ). We shall use Ψ 0 s (∆, K; η, x) to denote the set of η-equidistributed nilsequences in Ψ s (∆, K; η, x) satisfying the additional condition that G/Γ F = 0.…”

“…In this section, we collect Bombieri-Vinogradov type estimates for nilsequences from [28] that we shall need. Theorems 7.1 and 7.2 below are slight generalizations of [28,Theorem 4.3] and [28,Theorem 4.4], respectively. Theorem 7.1.…”

“…functions θ 1 and θ 2 . This will follow by working out a reduction to the case of equidistributed higher order Bohr sets (Section 5) and then applying weighted versions of Chen's and Maynard's sieves (Section 6), and adapting a Bombieri-Vinogradov theorem for primes twisted by nilsequences [28], proven by the last two authors. We also note that our transference principle is slightly stronger than the Green-Tao transference principle even for translation-invariant systems in the sense that our pseudorandomness requirement is weaker (we do not need the correlation condition from [9]); we achieve this relaxation by applying the work of Dodos and Kanellopoulos [5].…”