Quantitative uniform distribution results for geometric progressions

By a classical theorem of Koksma the sequence of fractional parts $(\{x^n\}{)}_{n\ge 1}$ 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 $(\{\xi x^{s_n}\}{)}_{n\ge 1}$ 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.