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Mahler’s question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory

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Abstract

In this paper, we consider Mahler’s question for intrinsic Diophantine approximation on the triadic Cantor set \(\mathcal {K}\), i.e., approximating the points in \(\mathcal {K}\) by rational numbers inside \(\mathcal {K}\):

$$\begin{aligned} {\mathcal {W}}_{\mathcal {K}}(\psi ):=\left\{ x\in \mathcal {K}: |x-p/q|<\psi (q), \, {\text {for infinitely many}}\ p/q\in \mathcal {K}\right\} . \end{aligned}$$

By using the intrinsic denominator \(q_{{\text {int}}}\) instead of the regular denominator q of a rational \(p/q\in \mathcal {K}\) in \(\psi (\cdot )\), we present a complete metric theory for this variant of the set \(\mathcal {W}_{\mathcal {K}}(\psi )\), which yields a divergence theory of Mahler’s question.

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Acknowledgements

The authors show their sincere appreciations to Professor Teturo Kamae (Osaka City University) on the discussion of the proof of Lemma 4.1, and the anonymous referee for extremely careful reading and helpful suggestions. This work is supported by NSFC of China (Grant Nos. 12171172 and 12331005).

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  1. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, 430074, China

    Bo Tan, Baowei Wang & Jun Wu

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  1. Bo Tan
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  3. Jun Wu
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Correspondence to Baowei Wang.

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Tan, B., Wang, B. & Wu, J. Mahler’s question for intrinsic Diophantine approximation on triadic Cantor set: the divergence theory. Math. Z. 306, 2 (2024). https://doi.org/10.1007/s00209-023-03397-1

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