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

We extend to Segal–Piatetski-Shapiro sequences previous results on the Luca–Schinzel question. Namely, we prove that for any real c larger than 1, the sequence ( ∑ m ≤ n φ ( ⌊ m c ⌋ ) / ⌊ m c ⌋ ) n ≥ 1 is dense modulo 1, where φ 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 ⌊ m c ⌋ modulo R contains any block of consecutive residues of a given length.