Subspaces of Z^n or R^n having Dimension (n-\varepsilon) in the (n-\varepsilon)-Expansion

Abstract

In the following we construct spaces of dimension $(n\pm \varepsilon)$ lying in the neighborhood of $\mathbb{Z}^n, \mathbb{Z}^n$ in the context of the $(n-\varepsilon)$-expansion. We provide means and criteria to deform the spaces of integer dimension into this neighborhood. We argue that the field theoretic models living on these deformed spaces are the continuation of the models defined on the corresponding integer valued spaces. Furthermore we perform the continuum limit of subgraphs of $\mathbb{Z}^n$ having non-integer dimension to the corresponding (fractal) subspaces of $\mathbb{Z}^n$. We make sense of a fractal volume measure like $d^{(n-\varepsilon)}x$.