A note on logarithmic equidistribution

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.