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Vinogradov three prime theorem with Piatetski-Shapiro primes

Yu-Chen Sun, Shanshan Du, Hao Pan

Abstract

We prove that, for any $c_1,c_2,c_3\in(1,41/35)$, every sufficiently large odd number $N$ can be represented as the sum of three primes $N = p_1 + p_2 +p_3$ such that $p_i = \lfloor n_{i}^{c_i}\rfloor$ for some $n_i \in{\mathbb N}$ for each $1 \leq i \leq 3$. Our arguments are based on a variant of Green's transference principle due to Matomäki, Maynard and Shao. We prove a necessary restriction estimate using Bourgain's strategy and employ Harman's sieve method to optimize our upper bound for $c_i$.

Vinogradov three prime theorem with Piatetski-Shapiro primes

Abstract

We prove that, for any , every sufficiently large odd number can be represented as the sum of three primes such that for some for each . Our arguments are based on a variant of Green's transference principle due to Matomäki, Maynard and Shao. We prove a necessary restriction estimate using Bourgain's strategy and employ Harman's sieve method to optimize our upper bound for .

Paper Structure

This paper contains 10 sections, 26 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Outline
  4. restriction estimates
  5. Lower bound sieve
  6. Upper bound Sieve
  7. Upper bound Fourier transform
  8. Minor arc case
  9. major arc case
  10. Roth's theorem for Piateski-Shapiro primes

Key Result

Theorem \oldthetheorem

For any $c_1,c_2,c_3\in(1,41/35)$, every sufficiently large odd $N$ can be represented as where $p_1,p_2,p_3$ are primes and $p_i\in\mathbb N^{c_i}$ for each $1\leq i\leq 3$.

Theorems & Definitions (58)

  • Definition 1: weak Balog-Friedlander condition
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark 1
  • Theorem \oldthetheorem
  • Theorem \oldthetheorem
  • Remark 2
  • Corollary \oldthetheorem
  • proof
  • proof : Proof of Theorem \ref{['AsyThm']}
  • ...and 48 more