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 \( (\alpha f(q_n))_{n \geq n_0} \) is uniformly distributed modulo one for any nonzero real number \( \alpha \). This generalises some former results due to Too, Goto and Kano where instead of \( (q_n)_n \) the sequence of primes was considered.