Packing intermediate dimensions of measures and applications
Abstract
We introduce a spectrum of dimensions of measures, the \emph{packing intermediate dimensions} dimP,θμ, θ∈(0,1], of a Borel probability measure μ on ℝn and the \emph{quasi-Hausdorff dimension} dimqHμ. We show that this spectrum encodes the information on μ that governs the packing dimension of its projections and Hölder images. First, dimqHμ is the exact threshold for the packing dimension profiles of Falconer and Howroyd: for every real 0<s≤ n, dimPsμ=s if and only if s≤dimqHμ. Second, we prove a lower bound for the profiles in terms of dimHμ, dimP,θμ and the Assouad dimension of suppμ, which improves the bound of Falconer and Mattila, the best previously known general lower bound for the packing dimension of projected measures. The improvement is strict whenever dimHμ<dimPμ and dimA(suppμ)<n, and when dimA(suppμ)=n it occurs exactly when dimP,θμ exceeds an explicit universal lower bound for some θ in an explicit range. We present examples of measures with dimHμ<dimPμ<dimA(suppμ)=n for which our bound gives the exact value dimPμV=min{dimPμ,m} for almost every m-dimensional subspace V, whereas the bound of Falconer and Mattila is strictly smaller. As applications, for a large family of random α-Hölder maps fω:ℝn→ℝm, we prove that, for compactly supported μ, dimPμfω=m almost surely if and only if αm≤dimqHμ, and that the packing analogue of Kahane's formula holds for fractional Brownian motions if and only if dimqHμ=dimPμ. Analogous results are obtained for analytic sets.
Related
- Sharp Projection Bounds for Intermediate Dimensions of Measures
- Fractal dimensions and quasisymmetric packing-minimalities of some homogeneous Moran sets
- Null-additive sets and packing measures
- Improved lattice-packing bounds in small dimensions
- The packing spectrum: interpolation between Hausdorff and packing dimensions
BibTeX
@misc{angelini2026packing,
title = {{Packing intermediate dimensions of measures and applications}},
author = {Nicolas Angelini},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.18929},
eprint = {2610.18929},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.18929},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Packing intermediate dimensions of measures and applications
AU - Nicolas Angelini
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.18929
UR - https://arxiv.org/abs/2610.18929
AB - We introduce a spectrum of dimensions of measures, the \emph{packing intermediate dimensions} dim_P,θμ, θ∈(0,1], of a Borel probability measure μ on ℝ^n and the \emph{quasi-Hausdorff dimension} dim_qHμ. We show that this spectrum encodes the information on μ that governs the packing dimension of its projections and Hölder images. First, dim_qHμ is the exact threshold for the packing dimension profiles of Falconer and Howroyd: for every real 0<s≤ n, dim_P^sμ=s if and only if s≤dim_qHμ. Second, we prove a lower bound for the profiles in terms of dim_Hμ, dim_P,θμ and the Assouad dimension of suppμ, which improves the bound of Falconer and Mattila, the best previously known general lower bound for the packing dimension of projected measures. The improvement is strict whenever dim_Hμ<dim_Pμ and dim_A(suppμ)<n, and when dim_A(suppμ)=n it occurs exactly when dim_P,θμ exceeds an explicit universal lower bound for some θ in an explicit range. We present examples of measures with dim_Hμ<dim_Pμ<dim_A(suppμ)=n for which our bound gives the exact value dim_Pμ_V=min{dim_Pμ,m} for almost every m-dimensional subspace V, whereas the bound of Falconer and Mattila is strictly smaller. As applications, for a large family of random α-Hölder maps f_ω:ℝ^n→ℝ^m, we prove that, for compactly supported μ, dim_Pμ_f_ω=m almost surely if and only if αm≤dim_qHμ, and that the packing analogue of Kahane's formula holds for fractional Brownian motions if and only if dim_qHμ=dim_Pμ. Analogous results are obtained for analytic sets.
ER -