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

Abstract

We prove that there are infinitely many solutions of $$|{\lambda }_0+{\lambda }_{1}p+{\lambda }_2{P}_r|<p^{-\tau },$$ where $r=3,$ $\tau =\frac{1}{118}$, and ${\lambda }_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with ${\lambda }_2\ne 0$ 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 $\left(1,1+\frac{1}{149}\right].$

23 pages. Number Theory