On the distribution of $αp^2$ modulo one over primes of the form $[n^c]$
Let $[\, \cdot\,]$ be the floor function and $\|x\|$ denote the distance from $x$ to the nearest integer. In this paper we show that whenever $α$ is irrational and $β$ is real then for any fixed $\frac{13}{14}<γ<1$, there exist infinitely many prime numbers $p$ satisfying the inequality \begin{equation*} \|αp^2+β\|< p^{\frac{13-14γ}{29}+\varepsilon} \end{equation*} and such that $p=[n^{1/γ}]$.
math.NT↗