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

## TL;DR

This paper constructs and analyzes spaces with non-integer dimensions near integer lattices, providing deformation criteria and exploring their implications for field theories and fractal measures.

## Contribution

It introduces methods to deform integer-dimensional spaces into non-integer neighborhoods and connects these to field theoretic models and fractal measures.

## Key findings

- Spaces of dimension n±ε are constructed near Z^n and R^n.
- Deformation criteria for these spaces are established.
- Continuum limits of non-integer dimensional subgraphs are related to fractal subspaces.

## 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 $d^{(n-\varepsilon)}x$.