Equidistribution in the complex plane and self-similar measures

We establish the pointwise equidistribution of self-similar measures in the complex plane. Let, whose complex conjugate is not a divisor of β, and a finite subset. Let μ be a non-atomic self-similar measure with respect to the IFS. For, if α and β are relatively prime, then we show that the sequence is equidistributed modulo one for μ-almost everywhere. 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.