Piatetski-Shapiro sequences via Beatty sequences

Abstract

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 \[\sum_{n\leq x}\varphi\left(\lfloor n^c\rfloor\right)-\frac 1c\sum_{n\leq x^c}\varphi(n)n^{\frac 1c-1},\] 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 $1/2$.