A note on logarithmic equidistribution

This is the authors' abstract. We don't add key points for this paper.

For every algebraic number [Formula: see text] on the unit circle which is not a root of unity we prove the existence of a strict sequence of algebraic numbers whose height tends to zero, such that the averages of the evaluation of [Formula: see text] at the conjugates are essentially bounded from above by [Formula: see text]. This completes a characterization on functions [Formula: see text] initiated by Autissier and Baker–Masser, who cover the cases [Formula: see text] and [Formula: see text], respectively. Using the same ideas we also prove analogues in the [Formula: see text]-adic setting.