The $(\varphi , 1)$ rectifiable subsets of Euclidean space

Abstract:

In this paper the structure of a subset $E \subset {{\mathbf {R}}^n}$ with ${{\mathbf {H}}^1}(E) < \infty$ has been studied by examining its intersection with various translated positions of a smooth hypersurface B. The following result has been established: Let B be a proper $(n - 1)$ dimensional smooth submanifold of ${{\mathbf {R}}^n}$ with nonzero Gaussian curvature at every point. If $E \subset {{\mathbf {R}}^n}$ with ${{\mathbf {H}}^1}(E) < \infty$, then there exists a countably 1-rectifiable Borel subset R of ${{\mathbf {R}}^n}$ such that $(E \sim R)$ is purely $({{\mathbf {H}}^1},1)$ unrectifiable and $(E \sim R) \cap (g + B) = \emptyset$ for almost all $g \in {{\mathbf {R}}^n}$. Furthermore, if in addition E is ${{\mathbf {H}}^1}$ measurable and $E \cap (g + B) = \emptyset$ for ${{\mathbf {H}}^n}$ almost all $g \in {{\mathbf {R}}^n}$ then ${{\mathbf {H}}^1}(E \cap R) = 0$. Consequently, E is purely $({{\mathbf {H}}^1},1)$ unrectifiable.