Warren's argument (which might or might not be valid -- it sounds good to me but I don't work in that area) reminds me of something I was told (but never understood) 30 years ago: that the fractal dimension of the Cartesian product of two sets need not be equal to the sum of the fractal dimensions of the two sets. Can anyone explain this to me? Jim
--ok, first of all, (A) I am not taking the cartesian product of arbitrary sets -- one of my sets is a line segment orthogonal to the other (2D) set. For that kind of Cartesian product I claim the Haussdorf dimension DOES sum. (B) Second, Jim Propp says that more generally, it need not sum. And why is that? Well, what IS "Haussdorf dimension"? It is this. Let your set S be "grown" to set S(r) which is all points within distance r away from S. Consider the measure of S(r) as a function of r>0. If S(r) is proportional to r^P in the limit r-->0+ (or merely bounded between c1*r^P and c2*r^P for some constants 0<c1<c2) then we'll say S has Haussdorf codimension=P i.e. H = HaussdorfDim(S) = D-P, where the universe we work within is D dimensional. By the way, note that Haussdorf dimension need not exist. So if S is a point (of finite collection of points) then P=D so H=0, that worked. If S is a line segment (or finite collection of curve segments) H=1, worked again. Etc. It works in the usual cases, but can be applied to goofy fractals as well. Good. Now it should be obvious to you (or you should be able to prove easily) that (A) was true. Actually whenever at least one of the two sets is a "well behaved" set (hence with integer dimension) then I claim the sum rule holds. Now: can we construct two fractals whose Cartesian product does NOT have summed Haussdorf dimension? This paper discusses... Kaoru HATANO: Notes on Hausdorff Dimensions of Cartesian Product Sets, HIROSHIMA MATH. J. 1 (1971) 17-25 http://projecteuclid.org/download/pdf_1/euclid.hmj/1206138139 I don't understand this paper, but it cites others, and it at start claims that (i) HaussdorfDim(A)+HaussdorfDim(B) <= HaussdorfDim(AxB) and also that (ii) max{ HaussdorfCoDim(A), HaussdorfCoDim(B) } <= HaussdorfCoDim(AxB). I think the basic intuition for why (i) is not an equality in general, is that the product set exhibits DIFFERENT power law scaling laws in different directions. Further the power can be different in different places -- Haussdorf sort of only takes something like an average or max whenever you've got a fractal like that, yielding only one number (assuming it succeeds in producing a number at all). If you always had the same power no matter which direction, then things would be easy, but no. Indeed, it is not obvious to me that the Haussdorf dimension of a Cartesian product set even necessarily exists at all, even when it does for both the two original sets. Hatano seems to take it for granted it does exist... which hopefully is because somebody had proved that earlier... but he does not cite a proof of that, nor even explicitly state that he is assuming this (but he is).