A threshold for full packing dimension of Hölder images of sets and measures
Abstract
We study when the image of a measure under a random Hölder map attains full packing dimension. Our main tool is a family of packing intermediate dimension profiles dimP,θsμ, indexed by θ∈(0,1] and s>0, which refine the packing dimension profiles of Falconer and Howroyd and reduce to them at θ=1. For a large family of random α-Hölder maps fω:ℝn→ℝm, which includes index-α fractional Brownian motion as a particular case, we prove that for every compactly supported Borel probability measure μ on ℝn, dimP μfω=m almost surely Longleftrightarrow αm≤ limθ→ 0dimP,θn μ. We further study the profiles dimPsμ themselves, obtaining a quantitative lower bound and a Marstrand-type identity for their limiting behavior as θ→ 0. Finally, we obtain the analogous characterization for analytic sets: dimP fω(E)=m almost surely Longleftrightarrow αm≤ limθ→ 0dimP,θn E, where dimP,θn E=sup{dimP,θn μ: μ∈ ℳc+(E) }.
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BibTeX
@misc{angelini2026threshold,
title = {{A threshold for full packing dimension of Hölder images of sets and measures}},
author = {Nicolas Angelini},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11301},
eprint = {2609.11301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11301},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A threshold for full packing dimension of Hölder images of sets and measures
AU - Nicolas Angelini
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11301
UR - https://arxiv.org/abs/2609.11301
AB - We study when the image of a measure under a random Hölder map attains full packing dimension. Our main tool is a family of packing intermediate dimension profiles dim_P,θ^sμ, indexed by θ∈(0,1] and s>0, which refine the packing dimension profiles of Falconer and Howroyd and reduce to them at θ=1. For a large family of random α-Hölder maps f_ω:ℝ^n→ℝ^m, which includes index-α fractional Brownian motion as a particular case, we prove that for every compactly supported Borel probability measure μ on ℝ^n, dim_P μ_f_ω=m almost surely Longleftrightarrow αm≤ lim_θ→ 0dim_P,θ^n μ. We further study the profiles dim_P^sμ themselves, obtaining a quantitative lower bound and a Marstrand-type identity for their limiting behavior as θ→ 0. Finally, we obtain the analogous characterization for analytic sets: dim_P f_ω(E)=m almost surely Longleftrightarrow αm≤ lim_θ→ 0dim_P,θ^n E, where dim_P,θ^n E=sup{dim_P,θ^n μ: μ∈ ℳ_c^+(E) }.
ER -