We supplement the well known upper and lower box-counting product inequalities to give the new product formula {align*} LBF + LBG &≤ LB(F× G) \\ &≤ min(LBF + BG,BF + LBG)\\ &≤ max(LBF + BG,BF + LBG)\\ & ≤ B(F× G)\\ &≤ BF+BG {align*} 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.
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Robinson et al. (2013) studied this question.