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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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The $(\varphi , 1)$ rectifiable subsets of Euclidean space
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by Samir Kar PDF
Trans. Amer. Math. Soc. 237 (1978), 353-371 Request permission

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.
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Additional Information
  • © Copyright 1978 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 237 (1978), 353-371
  • MSC: Primary 49F20
  • DOI: https://doi.org/10.1090/S0002-9947-1978-0487725-X
  • MathSciNet review: 0487725