Some geometric properties of sets defined by digit restrictions

Let ES,D be a set defined by digit restrictions. In this paper, we determine the Assouad dimension of ES,D. As an application, we construct a set of different Hausdorff, lower box, upper box, and Assouad dimensions. Also, we prove that the set ES,D is either quasisymmetrically thick or thin. In addition, we show that ES,D is quasisymmetrically minimal provided that dimHES,D = 1.