Synchronizing automatic sequences along Piatetski-Shapiro sequences

The purpose of this paper is to study subsequences of synchronizing k-automatic sequences (a(n))n≥0 along Piatetski-Shapiro Sequences ⌊nc⌋ with c>1. In particular we show that (a(⌊nc⌋))n≥0 satisfies a prime number theorem of the form ∑n≤xΛ(n)a(⌊nc⌋)∼Cx, and, furthermore, that it is deterministic. As an interesting additional result we show that the sequence (⌊nc⌋ mod m)n≥0 has polynomial subword complexity.