Distributions of finite sequences represented by polynomials in Piatetski-Shapiro sequences.

By using the work of Frantzikinakis and Wierdl, we can see that for all d ∈ N , α ∈ (d , d + 1) , and integers k ≥ d + 2 and r ≥ 1 , there exist infinitely many n ∈ N such that the sequence (⌊ (n + r j) α ⌋) j = 0 k − 1 is represented as ⌊ (n + r j) α ⌋ = p (j) , j = 0 , 1 , ... , k − 1 , by using some polynomial p (x) ∈ Q [ x ] of degree at most d. In particular, the above sequence is an arithmetic progression when d = 1. In this paper, we show the asymptotic density of such numbers n as above. When d = 1 , the asymptotic density is equal to 1 / (k − 1). Although the common difference r is arbitrarily fixed in the above result, we also examine the case when r is not fixed. Most results in this paper are generalized by using functions belonging to Hardy fields. [ABSTRACT FROM AUTHOR]