THREE-TERM ARITHMETIC PROGRESSIONS OF PIATETSKI-SHAPIRO SEQUENCES (Problems and Prospects in Analytic Number Theory)

Abstract

For every non-integral α > 1, the sequence of the integer parts of n^α (n = 1, 2, ... ) is called the Piatetski-Shapiro sequence with exponent a. Let PS(α) be the set of all those terms. In a previous study, Matsusaka and the author studied the set of α ∈ I such that PS(α) contains infinitely many arithmetic progressions of length 3, where I is a closed interval of [2, ∞). As a corollary of their main result , they showed that the set is uncountable and dense in I. The aim of this article is to provide a direct proof of this result.