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Hausdorff and Packing Measures for β-L{ü}roth Infinite Iterated Function Systems

  • Junjue Miao
  • Tianrui Wang
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Abstract

We study the Hausdorff and packing measures of attractors generated by infinite β-Lüroth iterated function systems. Let I⊂ℕ be infinite and let dimH EI=h. For a regular system, we first give conditions for the positivity of ℋh(EI) and the finiteness of 𝒫h(EI) in terms of first-level Lüroth intervals. We then introduce tail-regular partitions and the scale Rn=Bn∞/Bn. Under tail-regularity, the Hausdorff and packing conditions reduce to counting conditions for I on intervals of length comparable to Rn, together with a tail condition in the packing case. Finally, we apply these results to regularly varying partitions and stretched-exponential partitions.

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Cite

BibTeXRIS
BibTeX
@misc{miao2026hausdorff,
  title = {{Hausdorff and Packing Measures for $β$-Lüroth Infinite Iterated Function Systems}},
  author = {Junjue Miao and Tianrui Wang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.34076},
  eprint = {2610.34076},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.34076},
}
RIS
TY  - UNPB
TI  - Hausdorff and Packing Measures for β-L{ü}roth Infinite Iterated Function Systems
AU  - Junjue Miao
AU  - Tianrui Wang
PY  - 2026
DA  - 2026/10/25
PB  - arXiv
DO  - 10.48550/arxiv.2610.34076
UR  - https://arxiv.org/abs/2610.34076
AB  - We study the Hausdorff and packing measures of attractors generated by infinite β-Lüroth iterated function systems. Let I⊂ℕ be infinite and let dim_H E_I=h. For a regular system, we first give conditions for the positivity of ℋ^h(E_I) and the finiteness of 𝒫^h(E_I) in terms of first-level Lüroth intervals. We then introduce tail-regular partitions and the scale R_n=B_n^∞/B_n. Under tail-regularity, the Hausdorff and packing conditions reduce to counting conditions for I on intervals of length comparable to R_n, together with a tail condition in the packing case. Finally, we apply these results to regularly varying partitions and stretched-exponential partitions.
ER  -