Borel subrings of the reals

Edgar, G.; Miller, Chris

doi:10.1090/S0002-9939-02-06653-4

Borel subrings of the reals
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by G. A. Edgar and Chris Miller PDF

Proc. Amer. Math. Soc. 131 (2003), 1121-1129 Request permission

Abstract:

A Borel (or even analytic) subring of $\mathbb R$ either has Hausdorff dimension $0$ or is all of $\mathbb R$. Extensions of the method of proof yield (among other things) that any analytic subring of $\mathbb C$ having positive Hausdorff dimension is equal to either $\mathbb R$ or $\mathbb C$.

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