Dimension of homogeneous iterated function systems with algebraic translations

Abstract Let be the self‐similar measure associated with a homogeneous iterated function system on and a probability vector , where and . Recently by modifying the arguments of Varjú in [28], Rapaport and Varjú [24] showed that if are rational numbers and , then unless has exact overlaps. In this paper, we further show that the above result holds in the case when are algebraic numbers and . This is done by adapting the ideas and extending the results of Breuillard, Rapaport, and Varjú.