On a Diophantine inequality involving a prime and an almost-prime

TL;DR

This paper proves the existence of infinitely many solutions to a Diophantine inequality involving a prime and an almost-prime, with improved bounds and specific prime forms, advancing previous results in the field.

Contribution

It establishes new bounds for solutions to a Diophantine inequality involving primes and almost-primes, including Piatetski-Shapiro primes, improving prior work by Harman.

Findings

Infinitely many solutions for the inequality with prime p and almost-prime P_r.

Extension to primes of the form loor{n^c} with explicit bounds.

Improved approximation exponent au and broader prime conditions.

Abstract

We prove that there are infinitely many solutions of $$ |\lambda_0+\lambda_1p+\lambda_2P_r|<p^{-\tau}, $$ where $r=3,$ $\tau=\frac1{118}$, and $\lambda_0$ is an arbitrary real number and $\lambda_1,\lambda_2\in\BR$ with $\lambda_2\neq0$ and $0>\frac{\lambda_1}{\lambda_2}$ not in $\mathbb{Q}$. This improves a result by Harman. Moreover, we show that one can require the prime $p$ to be of the form $\floor{n^c}$ for some positive integer $n$, i.e. $p$ is a Piatetski-Shapiro prime, with $r=13$ and $\tau=\rho(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$