SOME REMARKS ON THE DIMENSIONS OF THE PRODUCTS OF CANTOR SETS

Using the properties of the concave function, we show that the Hausdor？？ dimension of the product Ca+b 2 ; a+b 2&pound;Ca+b 2 ; a+b 2 of the same symmetric Cantor sets is greater than that of the prod-uct Ca;b &pound; 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 prod-uct Ca;a &pound; Cb;b of the di？？erent symmetric Cantor sets is greater than that of the product Ca+b 2 ; a+b 2 &pound; Ca+b 2 ; a+b of the same sym- 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.