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

Frank Gilson

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

Number Theory 2026-01-27 v2

Authors: Frank Gilson

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 $\kappa_K>0$ .

Keywords

quadratic irrationals Cantor set Diophantine approximation exit time distance to Cantor set

Cite

@article{arxiv.2601.11799,
  title  = {Explicit separation of quadratic irrationals from the middle-third Cantor set},
  author = {Frank Gilson},
  journal= {arXiv preprint arXiv:2601.11799},
  year   = {2026}
}

Comments

22 pages, main result is conditional, unconditional proof requires substantial work beyond this initial research

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

Abstract

Keywords

Cite

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