Squares in Piatetski--Shapiro Sequences

We study the distribution of squares in a Piatetski-Shapiro sequence ${(\lfloor n^c\rfloor )}_{n\in \mathbb{N}}$ with $c>1$ and $c\in ̸\mathbb{N}$. We also study more general equations $\lfloor{n^c}\rfloor = sm^2$,$n,m\in \mathbb N$,$1\le n \le N$foraninteger$s$ and obtain several bounds on the number of solutions for a fixed $s$ and on average over $s$ in an interval. These results are based on various techniques chosen depending on the range of the parameters.