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

TL;DR

This paper establishes bounds on how closely quadratic irrationals can be separated from the middle-third Cantor set, under certain conditions, revealing limitations on their proximity.

Contribution

It introduces new bounds on the distance between quadratic irrationals and the Cantor set, under mild assumptions, extending understanding of their distribution.

Findings

Bound on exit times of quadratic irrationals from the Cantor set

Lower bounds on the distance between quadratic irrationals and the Cantor set

Conditions under which these bounds hold

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 $\kappa_K>0$.