Equidistribution in the complex plane and self-similar measures

Source abstract

Abstract We establish the pointwise equidistribution of self-similar measures in the complex plane. Let $\beta \in \mathbb Z[\mathrm{i}]$ , whose complex conjugate $\overline{\beta}$ is not a divisor of β , and $T \subset \mathbb Z[\mathrm{i}]$ a finite subset. Let µ be a non-atomic self-similar measure with respect to the IFS $\big\{f_{t}(z)=\frac{z+t}{\beta}\colon t\in T\big\}$ . For $\alpha \in \mathbb Z[\mathrm{i}]$ , if α and β are relatively prime, then we show that the sequence $(\alpha^n z)_{n\ge 1}$ is equidistributed modulo one for µ -almost everywhere $z \in \mathbb{C}$ . 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.