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

Abstract

An earlier form of this manuscript claimed, subject to a ``shallow contribution'' hypothesis, a log-square upper bound for the first occurrence of the ternary digit $1$ in a quadratic irrational, and consequently a quantitative separation from the middle-third Cantor set. Those conclusions are withdrawn. The argument used a decomposition defined only after finite exit had been assumed in order to prove finite exit, and the proposed quadratic-field-uniform Thue--Mahler estimate did not follow from the cited $S$-unit theory: the relevant set $S$ varies with the minimal polynomial, and the displayed $S$-unit quotient does not retain the power-of-$3$ exponent without an additional normalization. This replacement records the defects and retains the unconditional part of the argument. For a quadratic irrational whose orbit exits, we prove the exact exit-to-distance identity and an elementary height-dependent clearance bound. We also give a corrected universal bound for a single $L$-run and show that an exceptionally long $R$-to-$L$ transition produces a bounded value of a binary quadratic norm form at a power of $3$. A finite-prefix decomposition identifies the additional uniform orbit-counting theorem that would be required for a genuine separation result. No assertion that a quadratic irrational lies outside the Cantor set is made here.

arXiv commentMajor correction. The earlier log-square exit-time theorem and the resulting separation bound are withdrawn. The proof assumed finite exit while using the hypothesis to prove finite exit, and its field-uniform S-unit/Thue--Mahler estimate was not established. This replacement records the gaps and retains only the unconditional dynamical, integrality, and bounded-norm reductions