Piatetski-Shapiro sequences via Beatty sequences

Integer sequences of the form $\lfloor n^c\rfloor$, where $1< c< 2$, can be locally approximated by sequences of the form $\lfloor n\alpha+\beta\rfloor$ in a very good way. Following this approach, we are led to an estimate of the difference \[ \sum_{n\leq x}\varphi(\lfloor n^c\rfloor) - \frac 1c\sum_{n\leq x^c}\varphi(n)n^{1/c-1}, \] which measures the deviation of the mean value of $\varphi$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1< c\leq 1.42$ the subsequence of the Thue–Morse sequence indexed by $\lfloor n^c\rfloor$ attains both of its values with asymptotic density $1/2$.