  By a classical theorem of Koksma the sequence of fractional parts
$(\{x^n\})_{n \geq 1}$ is uniformly distributed for almost all values of $x$.
In the present paper we obtain an exact quantitative version of Koksma's
theorem, by calculating the precise asymptotic order of the discrepancy of
$(\{\xi x^{s_n}\})_{n \geq 1}$ for typical values of $x>1$ (in the sense of
Lebesgue measure). Here $\xi>0$ is an arbitrary constant, and $(s_n)_{n \geq
1}$ can be any increasing sequence of positive integers.
