Uniform distribution of $\alpha n$ modulo one for a family of integer sequences

We show that the sequence $(\alpha n{)}_{n\in \mathcal{B}}$ is uniformly distributed modulo 1, for every irrational $\alpha$, provided $\mathcal{B}$ belongs to a certain family of integer sequences, which includes the prime, almost prime, squarefree, practical, densely divisible and lexicographical numbers. We also give an estimate for the discrepancy if $\alpha$ has finite irrationality measure.