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If $w=(q_k{)}_{k\in \mathbf{N}}$ denotes the sequence of best approximation denominators to a real $\alpha$, and $s_{\alpha }\left(n\right)$ denotes the sum of digits of $n$ in the digit representation of $n$ to base $w$, then for all $x$ irrational, the sequence $\left(s_{\alpha }\right(n)·x{)}_{n\in \mathbf{N}}$ is uniformly distributed modulo one. Discrepancy estimates for the discrepancy of this sequence are given, which turn out to be best possible if $\alpha$ has bounded continued fraction coefficients.