A note on logarithmic equidistribution

Abstract

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

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