The packing spectrum: interpolation between Hausdorff and packing dimensions
Abstract
We introduce a family of metric outer measures whose critical exponents form a packing spectrum interpolating between Hausdorff and packing dimensions. For analytic sets, we characterize each spectral value as the supremum of the Hausdorff dimension of a product with a compact testing set minus the testing set's upper intermediate dimension at the complementary parameter. The associated product bounds refine the classical Hausdorff--packing inequalities and can give exact Hausdorff additivity when both factors have distinct endpoint dimensions. Below the packing endpoint, we prove a variational principle in terms of windowed local mass exponents and sharp parameter estimates implying local Lipschitz continuity; a jump at the endpoint is possible. We characterize all attainable profiles by monotonicity, continuity below the packing endpoint, and a sharp upper right Dini-derivative bound. Every admissible profile is realized by a compact binary digit set whose uniform measure attains the finite-window and classical packing variational suprema. Complementary digit positions yield reflected upper intermediate-dimension profiles and a single factor attaining the Hausdorff product bounds at every parameter, with an Ahlfors regular product measure.
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BibTeX
@misc{liu2026packing,
title = {{The packing spectrum: interpolation between Hausdorff and packing dimensions}},
author = {Peizhi Liu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10375},
eprint = {2610.10375},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10375},
}
RIS
TY - UNPB
TI - The packing spectrum: interpolation between Hausdorff and packing dimensions
AU - Peizhi Liu
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.10375
UR - https://arxiv.org/abs/2610.10375
AB - We introduce a family of metric outer measures whose critical exponents form a packing spectrum interpolating between Hausdorff and packing dimensions. For analytic sets, we characterize each spectral value as the supremum of the Hausdorff dimension of a product with a compact testing set minus the testing set's upper intermediate dimension at the complementary parameter. The associated product bounds refine the classical Hausdorff--packing inequalities and can give exact Hausdorff additivity when both factors have distinct endpoint dimensions. Below the packing endpoint, we prove a variational principle in terms of windowed local mass exponents and sharp parameter estimates implying local Lipschitz continuity; a jump at the endpoint is possible. We characterize all attainable profiles by monotonicity, continuity below the packing endpoint, and a sharp upper right Dini-derivative bound. Every admissible profile is realized by a compact binary digit set whose uniform measure attains the finite-window and classical packing variational suprema. Complementary digit positions yield reflected upper intermediate-dimension profiles and a single factor attaining the Hausdorff product bounds at every parameter, with an Ahlfors regular product measure.
ER -