Fractional parts of generalized polynomials at prime arguments

Abstract

Let f(X) be a monic polynomial of degree d≥2 with real coefficients. In this paper, we obtain an upper bound for the discrepancy of the sequence ([f(p)α]β) generated by the generalized polynomial [f(X)α]β, where p runs through the set of all primes and α, β are irrational numbers satisfying certain conditions. We also study the small fractional parts of the sequence ([f(p)α]β) and prove a result analogous to that of Madritsch and Tichy (Theorem 2.3, [13]).