Log-Like Functions and Uniform Distribution Modulo One

For a function f satisfying f (x) = o((log x) ^K), K > 0, and a sequence of numbers (q_n) _n, we prove by assuming several conditions on f that the sequence (αf (q_n)) _n≥n0 is uniformly distributed modulo one for any nonzero real number α. This generalises some former results due to Too, Goto and Kano where instead of (q_n) _n the sequence of primes was considered.