Ball codes: A coding characterization of Hausdorff and packing dimensions

@misc{miyabe2026ball,
  title = {{Ball codes: A coding characterization of Hausdorff and packing dimensions}},
  author = {Kenshi Miyabe},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2608.29561},
  eprint = {2608.29561},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.29561},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Ball codes: A coding characterization of Hausdorff and packing dimensions
AU  - Kenshi Miyabe
PY  - 2026
DA  - 2026/08/30
PB  - arXiv
DO  - 10.48550/arxiv.2608.29561
UR  - https://arxiv.org/abs/2608.29561
AB  - We prove a purely classical, coding-theoretic characterization of Hausdorff and packing dimensions in ℝ^n. A \emph{ball code} assigns names to closed balls of ℝ^n: it is a partial map from a prefix-free set of finite binary strings to balls. For every nonempty E⊆ℝ^n, the Hausdorff dimension of E is the minimum, over all ball codes, of the supremum over x∈ E of the lower asymptotic rate at which x can be described by names of balls containing it; the packing dimension is obtained in the same way from the upper rates. The Hausdorff characterization is a Euclidean counterpart of Ryabko's coding theorem for combinatorial sources. We obtain the packing characterization from the standard characterization of packing dimension by upper modified box-counting dimension, without encoding ℝ^n into a sequence space. We then derive the point-to-set principle of J.~Lutz and N.~Lutz by replacing a minimizing ball code with a rational one and storing the resulting countable codebook in an oracle.
ER  -