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 (Σm n ( mc ) / mc )n is dense modulo 1, where 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 mc modulo R contains blocks of consecutive values which are in an arithmetic progression.
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