Synchronizing automatic sequences along Piatetski-Shapiro sequences

The purpose of this paper is to study subsequences of synchronizing $k$-automatic sequences $a(n)$ along Piatetski-Shapiro sequences $\lfloor n^c\rfloor$ with non-integer $c>1$. In particular, we show that $a(\lfloor n^c\rfloor )$ satisfies a prime number theorem of the form ${\sum }_{n\le x}\mathrm{\Lambda }(n)a(\lfloor n^c\rfloor )\sim Cx$, and, furthermore, that it is deterministic for $c\in \mathbb{R}\setminus \mathbb{Z}$. As an interesting additional result, we show that the sequence $\lfloor n^c\rfloor m$ has polynomial subword complexity.