A hybrid of two theorems of Piatetski-Shapiro

TL;DR

This paper investigates the solvability of a Diophantine inequality involving sums of Piatetski-Shapiro primes of a certain index, extending classical results to a new class of primes defined by fractional powers.

Contribution

It introduces new results on the solubility of Diophantine inequalities using Piatetski-Shapiro primes of non-integer index, blending two theorems in this area.

Findings

Established conditions for the inequality's solvability in Piatetski-Shapiro primes.

Extended classical theorems to primes of fractional power form.

Provided bounds on the number of solutions under certain parameters.

Abstract

Let $c > 1$ and $0 < \gamma < 1$ be real, with $c \notin \mathbb N$. We study the solubility of the Diophantine inequality \[ \left| p_1^c + p_2^c + \dots + p_s^c - N \right| < \varepsilon \] in Piatetski-Shapiro primes $p_1, p_2, \dots, p_s$ of index $\gamma$---that is, primes of the form $p = \lfloor m^{1/\gamma} \rfloor$ for some $m \in \mathbb N$.