Explicit separation of quadratic irrationals from the middle-third Cantor set

Abstract

Assuming a mild non-degeneracy condition excluding very low-level Cantor endpoints, and assuming a counting/input hypothesis for the contribution of non-deep orbit indices, we show that for the quadratic field $K=\mathbb{Q}(\alpha)$ there exist constants $A_K,B_K>0$ such that \[ \mathrm{exit}(\alpha)\ \le\ A_K\,(\log_3 H)^2 + B_K. \] Consequently, $\mathrm{dist}(\alpha,\mathcal C)\ge H^{-\kappa_K\log H}$ for some $\kappa_K>0$.