FRACTAL DIMENSIONS of SETS DEFINED by DIGIT RESTRICTIONS in ℝ²

We introduce a class of sets defined by digit restrictions in ℝ2 and study its fractal dimensions. Let ES,D be a set defined by digit restrictions in ℝ2. We obtain the Hausdorff and lower box dimensions of ES,D. Under some condition, we gain the packing and upper box dimensions of ES,D. We get the Assouad dimension of ES,D and show that it is 2 if and only if ES,D contains arbitrarily large arithmetic patches. Under some conditions, we study the upper spectrum, quasi-Assouad dimension and Assouad spectrum of ES,D. Finally, we give an intermediate value property of fractal dimensions of the class of sets.