Abstract
IN fractal geometry, two classes of sets play important roles. One is the regular set (the set Hausdorff and packing dimension coincide), the other is the set whose Bouligand dimension exists. A natural question is how to measure "the size" of these sets mentioned above. In this note, by using category, we answer this question. The main result is Theorem 1. Suppose that E c Rd . We denote by dimHE, dim,E , dimB~ and hE , respectively, the Hausdorff dimension, packing dimension, upper Bouligand dimension and lower Bouligand dimension of E . If dimBE = &BE, we say that the Bouligand dimension of E exists. For the details of definitions and properties of the above dimensions, see reference [I]. Given e>O,let V,(E)= 1z-ERd:d(x,~)<~\ , where d(x,E)=inf{ 11 x - y 11 :y
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