The Bombieri-Vinogradov theorem for nilsequences

Abstract

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 $\Vert \alpha p^k\Vert <\epsilon$, 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 $q\le x^{\theta }$, for almost all $q$.

55 pages. Referee comments incorporated. Formatted using the Discrete Analysis style file

References in corpus (4)

• Vinogradov's three primes theorem with almost twin primes
• Bombieri-Vinogradov for multiplicative functions, and beyond the $x^{1/2}$-barrier
• Gowers norms of multiplicative functions in progressions on average
• Bounded gaps between primes in short intervals

Cited by in corpus (1)

• Higher uniformity of arithmetic functions in short intervals I. All intervals