In Invent. Math.126 (2000), pp. 513–545, we gave a proof of Lang's conjecture on Abelian varieties leading to an effective bound for the number of translates involved. We show here that the method can be extended to give a similar statement for the ‘Mordell–Lang plus Bogomolov’ theorem proven by B. Poonen and independently by S. Zhang. We deal in detail with tori for which effective results have been obtained by J.-H. Evertse and H. P. Schlickewei; we improve on these mainly by providing polynomial bounds in the degree instead of doubly exponential ones. We also state a theorem for Abelian varieties. In both cases the strategy of proof is based on the approach of Mumford and Vojta–Faltings–Bombieri together with an effective Bogomolov property and therefore does not rely on either equidistribution nor subspace theorem arguments.
