Strict inequality in the box-counting dimension product formulas

Abstract

We supplement the well known upper and lower box-counting prod-uct inequalities to give the new product formula dimLB F + dimLB G ≤ dimLB (F × G) ≤ min (dimLB F + dimB G; dimB F + dimLB G) ≤ max (dimLB F + dimB G; dimB F + dimLB G) ≤ dimB (F × G) ≤ dimB F + dimB G for subsets of metric spaces. We develop a procedure for constructing sets so that the upper and lower box-counting dimensions of these sets and their product can take arbitrary values satisfying the above product formula. In particular we illustrate how badly behaved both the lower and upper box-counting dimensions can be on taking products.