$k$th powers in a generalization of Piatetski-Shapiro sequences

The article considers a generalization of Piatetski-Shapiro sequences in the sense of Beatty sequences. The sequence is defined by ${\left(\lfloor \alpha n^c+\beta \rfloor \right)}_{n=1}^{\mathrm{\infty }}$, where $\alpha \ge 1$, $c>1$, and $\beta$ are real numbers.

The focus of the study is on solving equations of the form $\lfloor \alpha n^c+\beta \rfloor =s{m}^k$, where $m$ and $n$ are positive integers, $1\le n\le N$, and $s$ is an integer. Bounds for the solutions are obtained for different values of the exponent $k$, and an average bound is derived over $k$-free numbers $s$ in a given interval.