SOME REMARKS ON THE DIMENSIONS OF THE PRODUCTS OF CANTOR SETS

Abstract

Using the properties of the concave function, we show that the Hausdorff dimension of the product C a+b 2 , a+b of the same symmetric Cantor sets is greater than that of the product Ca,b × Ca,b of the same anti-symmetric Cantor sets. Further, for 1/e2 < a, b < 1/2, we also show that the dimension of the product Ca,a × Cb,b of the different symmetric Cantor sets is greater than that of the product C a+b of the same sym2 , a+b 2 metric Cantor sets using the concavity. Finally we give a concrete example showing that the latter argument does not hold for all 0 < a, b < 1/2.

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