On a question of Luca and Schinzel over Segal-Piatetski-Shapiro sequences

Abstract

We extend to Segal-Piatetski-Shapiro sequences previous results on the Luca-Schinzel question over integral valued polynomial sequences. Namely, we prove that for any real $c$ larger than $1$ the sequence $({\sum }_{m\le n}\phi (\lfloor m^c\rfloor )/\lfloor m^c\rfloor {)}_n$ is dense modulo $1$, where $\phi$ denotes Euler's totient function. The main part of the proof consists in showing that when $R$ is a large integer, the sequence of the residues of $\lfloor m^c\rfloor$ modulo $R$ contains blocks of consecutive values which are in an arithmetic progression.

12 pages. Second version with minor revisions