Diophantine approximation with mixed powers of Piatetski-Shapiro primes

TL;DR

This paper proves the existence of infinitely many prime triples with mixed powers satisfying a Diophantine approximation inequality, under certain conditions on the primes and parameters, extending results in prime number theory.

Contribution

It establishes new results on Diophantine approximation involving primes of mixed powers, specifically Piatetski-Shapiro primes, with explicit bounds and conditions.

Findings

Infinitely many prime triples satisfy the approximation inequality.

Prime triples are of Piatetski-Shapiro type with fractional powers.

The results extend previous work on prime approximations with mixed powers.

Abstract

Let $[\,\cdot\,]$ denote the floor function. In this paper, we show that whenever $\eta$ is real and the constants $\lambda _i$ satisfy some necessary conditions, then for any fixed $\frac{63}{64}<\gamma<1$ and $\theta>0$, there exist infinitely many prime triples $p_1,\, p_2,\, p_3$ satisfying the inequality \begin{equation*} |\lambda _1p_1 + \lambda _2p_2 + \lambda _3p^2_3+\eta|<\big(\max \{p_1, p_2, p^2_3\}\big)^{{\frac{63-64\gamma}{52}}+\theta} \end{equation*} and such that $p_i=[n_i^{1/\gamma}]$, $i=1,\,2,\,3$.