Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Bombieri-Vinogradov theorem for nilsequences

Xuancheng Shao, Joni Teräväinen

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 $\|αp^k\|<\varepsilon$, 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 $q\leq x^θ$, for almost all $q$.

The Bombieri-Vinogradov theorem for nilsequences

Abstract

We establish results of Bombieri-Vinogradov type for the von Mangoldt function twisted by a nilsequence. In particular, we obtain Bombieri-Vinogradov type results for the von Mangoldt function twisted by any polynomial phase ; 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 obeying a "nil-Bohr set" condition, such as , 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 , for almost all .

Paper Structure

This paper contains 21 sections, 43 theorems, 237 equations.

Table of Contents

  1. Introduction
  2. Results for nilsequence twists
  3. Results for polynomial phase twists
  4. Notation
  5. Applications
  6. Bounded gaps between primes in Bohr sets
  7. Chen primes in Bohr sets
  8. Linear equations in primes in arithmetic progressions
  9. Future work
  10. Quantitative equidistribution of polynomial sequences
  11. Bombieri--Vinogradov theorems for equidistributed nilsequences
  12. Statements of results for equidistributed nilsequences
  13. Proof that Theorem \ref{['equidist-1/4']} implies Theorem \ref{['thm:1/4']}
  14. Proof that Theorems \ref{['equidist-1/3']} and \ref{['equidist-1/2']} imply Theorems \ref{['thm:1/3']} and \ref{['thm:1/2']}
  15. Type I estimates
  16. ...and 6 more sections

Key Result

Theorem 1.3

Let an integer $s\geq 1$, large real numbers $A, \Delta\geq 2$, and a small real number $\varepsilon\in (0,1/4)$ be given. Then for any $x \geq 2$, we have where $W = \mathscr{P}((\log x)^C)$ for some constant $C = C(A,s,\Delta,\varepsilon)$.

Theorems & Definitions (84)

  • Definition 1.1: Nilsequences
  • Definition 1.2: Bounded complexity nilsequences
  • Theorem 1.3
  • Theorem 1.4
  • Definition 1.5: Well-factorable sequences
  • Theorem 1.6
  • Remark 1.7
  • Remark 1.8
  • Corollary 1.9
  • Corollary 1.10
  • ...and 74 more