Category and dimension of compact subsets of R^n

IN fractal geometry, two classes of sets play important roles. One is the regular set (the setHausdorff and packing dimension coincide), the other is the set whose Bouligand dimensionexists. A natural question is how to measure 'the size' of these sets mentioned above. In thisnote, by using category, we answer this question. The main result is Theorem 1.