Equidistribution in the complex plane and self-similar measures

We establish the pointwise equidistribution of self-similar measures in the complex plane. Let β ∈ Z[i] , whose complex conjugate β̄ is not a divisor of β , and T ⊂ Z[i] a finite subset. Let µ be a non-atomic self-similar measure with respect to the IFS ₜ(z)=z+t/β t∈ T\ . For α ∈ Z[i] , if α and β are relatively prime, then we show that the sequence (αⁿ z)n≥ 1 is equidistributed modulo one for µ -almost everywhere z ∈ C . We also discuss normality of radix expansions in Gaussian integer base, and obtain pointwise normality. Our results generalize partially the classical results in the real line to the complex plane.