Hausdorff dimension and countable Borel equivalence relations

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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 .
Funding Information
  • National Science Foundation (2348788)
  • National Science Foundation (10.55776/P36781)
  • Austrian Science Fund (2348788)
  • Austrian Science Fund (10.55776/P36781)