Synchronizing automatic sequences along Piatetski-Shapiro sequences

Abstract

The purpose of this paper is to study subsequences of synchronizing k-automatic sequences a(n) along Piatetski-Shapiro sequences ⌊nc⌋ with non-integer c>1. In particular, we show that a(⌊n^c⌋) satisfies a prime number theorem of the form ∑_{n≤x} Λ(n)a(⌊n^c⌋)∼Cx, and, furthermore, that it is deterministic for c∈ℝ∖ℤ. As an interesting additional result, we show that the sequence ⌊nc⌋modm has polynomial subword complexity.