Approximation of adelic divisors and equidistribution of small points

We study the asymptotic distribution of the Galois orbits of generic sequences of algebraic points of small height in a projective variety over a number field. Our main result is a generalization of Yuan's equidistribution theorem that applies to heights for which Zhang's lower bound for the essential minimum is not necessarily an equality. It extends to all projective varieties a theorem of Burgos Gil, Philippon, Rivera-Letelier and the second author for toric varieties. It also applies to sums of canonical heights for an algebraic dynamical system, and in particular it recovers Kühne's semiabelian equidistribution theorem. We also generalize previous work of Chambert-Loir and Thuillier to obtain new logarithmic equidistribution results. Finally we extend our main result to the quasi-projective setting recently introduced by Yuan and Zhang.