The Bombieri-Vinogradov theorem for nilsequences

We establish results of Bombieri-Vinogradov type for the von Mangoldt
function $\Lambda(n)$ 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 $\|\alpha p^k\|<\varepsilon$, exhibit
bounded gaps. Secondly, we show that the Chen primes are well-distributed in
nil-Bohr sets, generalizing a result of Matom\"aki. Thirdly, we generalize the
Green-Tao result on linear equations in the primes to primes belonging to an
arithmetic progression to large modulus $q\leq x^{\theta}$, for almost all $q$.