THE MEAN UNCERTAINTY AND DISTANCE DIMENSION OF MULTIDIMENSIONAL CANTORIAN SPACES

Several ways of describing the internal structure of infinite point se ts and determining a corresponding average dimension are outlined. Pos sible relevance to discretisation of quantum space-time we are discuss ed. Finally, a method for determining the average internal distance be tween different Cantor points is proposed. The main conclusion is that a micro-scale Cantorian space-time may be based on an average Cantori an distance d(l)(0) congruent-to 0.629, while the smooth space-time is retrieved when d(l)(0) --> 0.5, which means d(l)(0) tends towards the average distance of a continuous smooth line at a vanishing resolutio n. This correspond to the case when space is viewed from very far, at the macro-scale of classical physics.