Internal cantor distance and entropy of multidimensional Peano-Hilbert spaces

Abstract Several approaches for characterizing the internal structure of infinite point sets and determining a corresponding average dimension are outlined. Possible relevance to the discretization problem of quantum space-time are discussed. Finally a method for determining the average internal distance between different Cantor points is proposed. Our main conclusion is that Cantorian space-time is based on an average Cantorian distance ≅ 0.629 while the smooth space-time is retrieved when → 0.5 which means d(O)c tends towards the average distance of a continuous smooth line at vanishing resolution when space is viewed from afar, at a very large scale.