A note on logarithmic equidistribution

For every algebraic number $\kappa$ 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 $f_{\kappa }(z)=log|z-\kappa |$ at the conjugates are essentially bounded from above by $-h(\kappa )$. This completes a characterization on functions $f_{\kappa }$ initiated by Autissier and Baker–Masser, who cover the cases $\kappa =2$ and $\left|\kappa \right|\ne 1$, respectively. Using the same ideas we also prove analogues in the $p$-adic setting.