Every analytic real closed proper sub-field of $\mathbb R$ has Hausdorff dimension zero. Equivalently, every analytic set of real numbers having positive Hausdorff dimension contains a transcendence base for $\mathbb R$.
"Hausdorff Dimension, Analytic Sets and Transcendence." Real Anal. Exchange 27 (1) 335 - 340, 2001/2002.