On the simultaneous distribution of powers of prime numbers modulo one

We study the distribution of the sequence of vectors \({(p^{{\alpha}_1},\ldots,p^{{\alpha}_k})}\) modulo one. Here p runs over prime numbers, k ≥ 2 is a fixed integer, α1, . . . , α k are fixed real numbers lying in the interval (s, s + 1) with s being a sufficiently large integer.