Work

Hausdorff dimension and countable Borel equivalence relations

Year: 2025

Type: article

Abstract: We show that if E E is a countable Borel equivalence relation on R n \mathbb {R}^n , then there is a closed subset A ⊆ [ 0 , 1 ] n A \subseteq [0,1]^n of Hausdorff dimension n n so that E ↾ A E \restriction A is smooth. More generally, if ≤ Q \leq _Q is a locally countable Borel quasi-order on 2 ω 2^\omega and g g is any gauge function of lower order than the identity, then there is a closed set A A so that A A is an antichain in ≤ Q \leq _Q and H g ( A ) > 0 H^g(A) > 0 .

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/17496

Authors: Andrew Marks, Dino Rossegger, Theodore Slaman

Institutions: University of California, Berkeley, TU Wien

Language: English

Topic: Advanced Topology and Set Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, American Institute of Mathematics, European Commission, Austrian Science Fund