Dimension of homogeneous iterated function systems with algebraic translations

Let μ be the self-similar measure associated with a homogeneous iterated function system Φ = \ λ x + tⱼ \ⱼ₌₁ᵐ on R and a probability vector (pⱼ)ⱼ₌₁ᵐ, where 0≠ λ∈ (-1,1) and tⱼ∈ R. Recently by modifying the arguments of Varj\'u (2019), Rapaport and Varj\'u (2024) showed that if t₁,…, tₘ are rational numbers and 0<λ<1, then μ =min \ 1, \; ∑ⱼ₌₁ᵐ pⱼlog pⱼ log |λ| \ unless Φ has exact overlaps. In this paper, we further show that the above equality holds in the case when t₁,…, tₘ are algebraic numbers and 0<|λ|<1. This is done by adapting and extending the ideas employed in the recent papers of Breuillard, Rapaport and Varj\'u.