Dimension of Bernoulli Convolutions in Rᵈ

For (λ₁,...,λd)=λ∈(0,1)ᵈ with λ₁>...>λd, denote by μλ the Bernoulli convolution associated to λ. That is, μλ is the distribution of the random vector ∑n≥0±(λ₁ⁿ,...,λdⁿ), where the ± signs are chosen independently and with equal weight. Assuming for each 1≤ j≤ d that λⱼ is not a root of a polynomial with coefficients ±1,0, we prove that the dimension of μλ equals min\ Lμλ,d\, where Lμλ is the Lyapunov dimension. More generally, we obtain this result in the context of homogeneous diagonal self-affine systems on Rᵈ with rational translations. The proof extends to higher dimensions the works of Breuillard and Varj\'u and Varj\'u regarding Bernoulli convolutions on the real line. The main novelty and contribution of the present work lies in an extension of an entropy increase result, due to Varj\'u, in which the amount of increase in entropy is given explicitly. The extension of this result to the higher-dimensional non-conformal case requires significant new ideas.