FRACTAL DIMENSIONS OF SETS DEFINED BY DIGIT RESTRICTIONS IN?R2

摘要

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

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