Quantitative uniform distribution results for geometric progressions

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