Extending Borel’s conjecture from measure to dimension
Year: 2026
Type: article
Abstract: We study subsets A A of R N {\mathbb {R}}^N , gauge functions f f and their associated Hausdorff measures H f H^f . A set A ⊆ R N A\subseteq {\mathbb {R}}^N has strong dimension f f if and only if H f > 0 H^f>0 and for all gauge functions h h of higher order than f f , H h ( A ) = 0 H^h(A)=0 . We show that it is consistent with the axioms of set theory that if A A has strong dimension f f then A A is σ \sigma -finite for H f H^f .
DOI: 10.1090/cams/65
Author: Theodore Slaman
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: hybrid
OpenAlex ID: W7159944225