Absolute continuity of projections and dimension conservation for self-similar sets and measures
Abstract
Let A⊂ℝd, d≥3, be the attractor of a self-similar IFS whose rotations generate a dense subgroup of SO(d). For every 1≤ k<dimH A and V∈Gr(d,k), we prove that πVA has positive k-dimensional Lebesgue measure uniformly in V, and dimH(A∩πV−1{y})=dimH A−k for Lebesgue-almost every y∈πVA. Moreover, if μ is a self-similar measure for such an IFS with It(μ)<∞ for some k<t<d, then every μV=(πV)*μ is equivalent to the k-dimensional Lebesgue measure restricted to πVA, with density fV satisfying supV∫{fV>M}fV dℒVk≲ e−c(log M)1/3. Under strong separation, the conditional measures on the fibres are exact dimensional of dimension dimHμ−k. The proof uses self-similarity to iterate L1 norms of differences of the Gaussian-smoothing of μV between scales. Unlike Lq, q>1, the L1 norm does not grow under rescaling, allowing Varjú's L2 weak gap theorem under dense support condition to be used, and avoiding to use the much stronger uniform spectral gap assumption.
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BibTeX
@misc{jin2026absolute,
title = {{Absolute continuity of projections and dimension conservation for self-similar sets and measures}},
author = {Xiong Jin and Tuomas Sahlsten},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29301},
eprint = {2609.29301},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29301},
}
RIS
TY - UNPB
TI - Absolute continuity of projections and dimension conservation for self-similar sets and measures
AU - Xiong Jin
AU - Tuomas Sahlsten
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.29301
UR - https://arxiv.org/abs/2609.29301
AB - Let A⊂ℝ^d, d≥3, be the attractor of a self-similar IFS whose rotations generate a dense subgroup of SO(d). For every 1≤ k<dim_H A and V∈Gr(d,k), we prove that π_VA has positive k-dimensional Lebesgue measure uniformly in V, and dim_H(A∩π_V^−1{y})=dim_H A−k for Lebesgue-almost every y∈π_VA. Moreover, if μ is a self-similar measure for such an IFS with I_t(μ)<∞ for some k<t<d, then every μ_V=(π_V)_*μ is equivalent to the k-dimensional Lebesgue measure restricted to π_VA, with density f_V satisfying sup_V∫_{f_V>M}f_V dℒ_V^k≲ e^−c(log M)^1/3. Under strong separation, the conditional measures on the fibres are exact dimensional of dimension dim_Hμ−k. The proof uses self-similarity to iterate L^1 norms of differences of the Gaussian-smoothing of μ_V between scales. Unlike L^q, q>1, the L^1 norm does not grow under rescaling, allowing Varjú's L^2 weak gap theorem under dense support condition to be used, and avoiding to use the much stronger uniform spectral gap assumption.
ER -