Small gaps between the Piatetski-Shapiro primes

TL;DR

This paper proves that for certain exponents, there are infinitely many intervals of consecutive Piatetski-Shapiro primes with arbitrarily many primes, highlighting small gaps between these special primes.

Contribution

It establishes the existence of infinitely many intervals with many Piatetski-Shapiro primes for exponents between 1 and 9/8, extending understanding of their distribution.

Findings

Infinitely many intervals contain at least m+1 primes for large enough k_0.

The result holds for exponents c between 1 and 9/8.

Demonstrates small gaps between Piatetski-Shapiro primes.

Abstract

Suppose that $1<c<9/8$. For any $m\geq 1$, there exist infinitely many $n$ such that $$ \{[n^c],\ [(n+1)^c],\ \ldots,\ [(n+k_0)^c]\} $$ contains at least $m+1$ primes, if $k_0$ is sufficiently large (only depending on $m$).