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A quantitative bound on Furstenberg-Sárközy patterns with shifted prime power common differences in primes

Mengdi Wang

Abstract

Let $k\geq1$ be a fixed integer, and $\mathcal P_N$ be the set of primes no more than $N$. We prove that if a set $\mathcal A\subset\mathcal P_N$ contains no patterns $p_1,p_1+(p_2-1)^k$, where $p_1,p_2$ are prime numbers, then \[ \frac{|\mathcal A|}{|\mathcal P_N|}\ll(\log\log N)^{-\frac{1}{4k^3+23k^2}}. \]

A quantitative bound on Furstenberg-Sárközy patterns with shifted prime power common differences in primes

Abstract

Let be a fixed integer, and be the set of primes no more than . We prove that if a set contains no patterns , where are prime numbers, then

Paper Structure

This paper contains 13 sections, 17 theorems, 233 equations.

Table of Contents

  1. Introduction
  2. Notations and Preliminaries
  3. Notations
  4. Preliminaries
  5. Exponential sum estimates
  6. Restriction for shifted prime powers
  7. A Local Inverse Theorem
  8. Density Increment
  9. Proof of Lemma \ref{['exceptional']}
  10. Restriction for prime numbers
  11. Estimation of the singular series (\ref{['singular-series']})
  12. Proof of Lemma \ref{['minorsd']}
  13. Conflict of interest statement

Key Result

Theorem 1.1

Let $k\geq1$ be a fixed integer, and $\mathscr P_N$ be the set of primes which are no more than $N$. If $\mathscr A\subset\mathscr P_N$ and $(p-1)^k\not\in \mathscr A-\mathscr A$ for all primes $p$, then

Theorems & Definitions (24)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1: Primes in short arithmetic progressions
  • Lemma 2.2: Exceptional pair result
  • Lemma 2.3: Alternative restriction
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2: Major-arc behavior of $S_d(\alpha)$
  • proof
  • ...and 14 more