Paper detail

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

We prove that there are infinitely many solutions of $$ |λ_0+λ_1p+λ_2P_r|<p^{-τ}, $$ where $r=3,$ $τ=\frac1{118}$, and $λ_0$ is an arbitrary real number and $λ_1,λ_2\in\BR$ with $λ_2\neq0$ and $0>\frac{λ_1}{λ_2}$ 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 $τ=ρ(c),$ a constant explicitly determined by $c$ supported in $\left(1, 1+\frac1{149}\right].$