A Zero--One Law for Subrings of the Reals Closed under Binary Interleaving

Abstract

Erdős Problem 1154 asks whether every number in $[0,1]$ occurs as the Hausdorff dimension of a subring or subfield of $\mathbb{R}$. We prove that a subring of $\mathbb{R}$ closed under binary interleaving has Hausdorff dimension either zero or one. Consequently, every Turing ideal and its associated subfield satisfy the same dichotomy.