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+rj)^α⌋)_j=0^k−1 is represented as ⌊(n+rj)^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.