REMARKS ON DIMENSIONS OF CARTESIAN PRODUCT SETS

Given metric spaces E and F, it is well known that dim(H) E + dim(H) F <= dim(H)( E x F) <= dim(H) E + dim(P) F, dim(H) E + dim(P) F <= dim(P)( E x F) <= dim(P) E + dim(P) F and dim(B)E + (dim) over bar F-B <= (dim) over bar (B)( E x F) <= (dim) over bar E-B + (dim) over bar F-B, where dim(H) E, dim(P) E, dim(B)E, (dim) over bar E-B denote the Hausdorff, packing, lower box-counting, and upper box-counting dimension of E, respectively. In this paper, we shall provide examples of compact sets showing that the dimension of the product E x F may attain any of the values permitted by the above inequalities. The proof will be based on a study on dimension of products of sets defined by digit restrictions.