Piatetski-Shapiro sequences via Beatty sequences

TL;DR

This paper studies Piatetski-Shapiro sequences, approximates them with Beatty sequences, and analyzes their mean values and distribution properties, including the behavior of the Thue-Morse sequence on these subsequences for specific exponents.

Contribution

It provides new estimates for the deviation of mean values of functions on Piatetski-Shapiro sequences and proves distribution results for the Thue-Morse sequence on these subsequences for certain exponents.

Findings

Derived an estimate for the difference in mean values involving exponential sums.

Proved that for 1<c≤1.42, the Thue-Morse sequence on the subsequence attains both values with density 1/2.

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 $1/2$.