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On the distribution of $αp$ modulo one over Piatetski-Shapiro primes

S. I. Dimitrov

TL;DR

This work studies the distribution of the linear form α p modulo 1 when p runs over Piatetski-Shapiro primes p=[n^{1/γ}] with γ in (11/12,1).The authors prove that there are infinitely many such primes for which ||α p + β|| is bounded by p^{(11-12γ)/26} log^6 p, for any irrational α and any real β.The proof adapts Vinogradov-type inequalities to the sparse Piatetski-Shapiro prime set using Vaughan-type decompositions and a smooth cutoff function F_Δ to isolate small fractional parts.The result advances understanding of equidistribution phenomena on thin prime subsequences and extends classical Vinogradov-type bounds to the Piatetski-Shapiro setting.

Abstract

Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denotes the distance from $x$ to the nearest integer. In this paper we show that whenever $α$ is irrational and $β$ is real then for any fixed $1<c<12/11$ there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|αp+β\|\ll p^{\frac{11c-12}{26c}}\log^6p \end{equation*} and such that $p=[n^c]$.

On the distribution of $αp$ modulo one over Piatetski-Shapiro primes

TL;DR

This work studies the distribution of the linear form α p modulo 1 when p runs over Piatetski-Shapiro primes p=[n^{1/γ}] with γ in (11/12,1).The authors prove that there are infinitely many such primes for which ||α p + β|| is bounded by p^{(11-12γ)/26} log^6 p, for any irrational α and any real β.The proof adapts Vinogradov-type inequalities to the sparse Piatetski-Shapiro prime set using Vaughan-type decompositions and a smooth cutoff function F_Δ to isolate small fractional parts.The result advances understanding of equidistribution phenomena on thin prime subsequences and extends classical Vinogradov-type bounds to the Piatetski-Shapiro setting.

Abstract

Let be the floor function and denotes the distance from to the nearest integer. In this paper we show that whenever is irrational and is real then for any fixed there exist infinitely many prime numbers satisfying the inequality \begin{equation*} \|αp+β\|\ll p^{\frac{11c-12}{26c}}\log^6p \end{equation*} and such that .

Paper Structure

This paper contains 7 sections, 7 theorems, 90 equations.

Table of Contents

  1. Introduction and statement of the result
  2. Notations
  3. Preliminary lemmas
  4. Proof of the theorem
  5. Outline of the proof
  6. Upper bound for $\mathbf{\Gamma}$
  7. The end of the proof

Key Result

Theorem 1

Let $\gamma$ be fixed with $11/12<\gamma<1$, $\alpha$ is irrational and $\beta$ is real. Then there exist infinitely many Piatetski-Shapiro prime numbers $p$ of type $\gamma$ such that

Theorems & Definitions (13)

  • Theorem 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • ...and 3 more