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 $$\sum _{n\le x}\phi (\lfloor n^c\rfloor )-\frac{1}{c}\sum _{n\le x^c}\phi (n)n^{\frac{1}{c}-1},$$ which measures the deviation of the mean value of $\phi$ on the subsequence $\lfloor n^c\rfloor$ from the expected value, by an expression involving exponential sums. As an application we prove that for $1<c\le 1.42$ the subsequence of the Thue-Morse sequence indexed by $\lfloor n^c\rfloor$ attains both of its values with asymptotic density $1\mathrm{/}2$.