Distance of Quadratic Algebraic Numbers from the Middle-Third Cantor Set
Frank Gilson
TL;DR
This work addresses Mahler's problem on whether the Cantor set $\mathcal{C}$ contains irrational algebraic numbers by establishing quantitative lower bounds for the distance from quadratic irrationals to $\mathcal{C}$. The authors encode each quadratic irrational's orbit under $\theta \mapsto \{3\theta\}$ via an exit time $\mathrm{exit}(\alpha)$ and a coarse boundary distance $\delta_{\mathcal{C}}(\alpha)$, proving unconditional bounds $\mathrm{exit}(\alpha)\le A(\log_3 H)^2+B$ and, consequently, $\mathrm{dist}(\alpha,\mathcal{C})\ge H^{-\kappa\log H}$ for some $\kappa>0$ when $\delta_{\mathcal{C}}(\alpha)\ge0.02$. In the generic case with $3\nmid a$ and $\delta_{\mathcal{C}}(\alpha)\ge0.02$, a stronger bound $\mathrm{exit}(\alpha)\le \log_3 H+1$ is obtained. The analysis blends dynamical methods (R/L-blocks and L-runs), Diophantine estimates (Liouville–Mahler), and Thue–Mahler theory, with computer-assisted interval checks used for finite-case verifications. The results yield the first broad unconditional lower bounds on $\mathrm{dist}(\alpha,\mathcal{C})$ for quadratic irrationals away from coarse boundary obstructions and open avenues toward higher-degree generalizations and Cantor-like sets.
Abstract
We study the distance from quadratic irrational numbers to the middle-third Cantor set $C$. Mahler asked whether $C$ contains any irrational algebraic numbers; this remains open even for quadratic irrationals. Rather than assuming an answer to this problem, we obtain uniform lower bounds for the distance from a quadratic irrational $α$ to $C$ in terms of the height $H$ of the minimal polynomial of $α$. We encode $α$ by its orbit under the map $x \mapsto 3x \bmod 1$ and define the exit time $\operatorname{exit}(α)$ as the first iterate that enters the middle interval $[1/3,2/3]$. Our main unconditional result is a quadratic exit bound $\operatorname{exit}(α) \le A (\log_3 H)^2 + B$ for absolute constants $A,B > 0$, valid for all quadratic irrationals whose orbit stays a fixed small distance away from the coarse Cantor boundaries. As a consequence we obtain a distance lower bound $\operatorname{dist}(α,C) \ge H^{-κ\log H}$ for some constant $κ> 0$. On the dynamical side we classify orbits by an $L/M/R$ coding and prove that the total number of visits to the right interval $[2/3,1)$ is $O(\log H)$. A finite case analysis on a bounded portion of the orbit is reduced to checking a finite list of explicit affine inequalities on subintervals of $[0,1]$, which we verify with short computer scripts; all Diophantine and dynamical estimates are proved by hand.