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

TL;DR

This paper proves that for any real number greater than 1, a certain sequence involving Euler's totient function and Segal-Piatetski-Shapiro sequences is dense modulo 1, extending previous results on polynomial sequences.

Contribution

It extends previous work on the Luca-Schinzel question to Segal-Piatetski-Shapiro sequences, showing density modulo 1 for a broad class of sequences involving fractional powers.

Findings

Sequence is dense modulo 1 for all real c > 1

Residues of sequences contain blocks in arithmetic progression

Main proof involves residue analysis of fractional power 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 $(\sum_{m\le n} \varphi(\lfloor m^c \rfloor) /\lfloor m^c \rfloor)_n$ is dense modulo $1$, where $\varphi$ 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.