Mahler's question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory
Abstract
In this paper, we consider the intrinsic Diophantine approximation on the triadic Cantor set $\mathcal{K}$, i.e. approximating the points in $\mathcal{K}$ by rational numbers inside $\mathcal{K}$, a question posed by K. Mahler. By using another height function of a rational number in $\mathcal{K}$, i.e. the denominator obtained from its periodic 3-adic expansion, a complete metric theory for this variant intrinsic Diophantine approximation is presented which yields the divergence theory of Mahler's original question.
- Publication:
-
arXiv e-prints
- Pub Date:
- February 2021
- arXiv:
- arXiv:2103.00544
- Bibcode:
- 2021arXiv210300544T
- Keywords:
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- Mathematics - Number Theory;
- 11J83;
- 11K55