QUANTITATIVE UNIFORM DISTRIBUTION RESULTS FOR GEOMETRIC PROGRESSIONS

摘要

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 ({ξx~sn})~(n≥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.