A CONJECTURE TO FIND FOUR-TERM ARITHMETIC PROGRESSIONS OF PIATETSKI-SHAPIRO SEQUENCES (Integrated Research on Random Dynamical Systems and Multi-Valued Dynamical Systems)

Abstract

For every non-integral a > 1, the sequence of the integer parts of n[a](n= 1, 2, ...) is called the Piatetski-Shapiro sequence with exponent a. Let PS(a) be the set of all terms of this sequence. The aim of this article is to propose a conjecture to find infinitely many four-term arithmetic progressions of PS(a).