PreprintCC0arXiv
On the Hausdorff measure of projections of self-similar sets
Abstract
For 0<d≤1/4, let 𝒞(d) be the four-corner Cantor set with Hausdorff dimension sd=log 4/log(1/d). Peres, Simon, and Solomyak, as well as Mattila, asked for which d the measure Hausd(pθ(𝒞(d))) is positive for almost every direction θ. It was open for the range 1/9≤ d≤1/6. In this paper, we show that, for δ< d≤1/6, there is a set IP(d) of positive Lebesgue measure such that for almost every θ∈IP(d), Hausd(pθ(𝒞(d)))=0, where δ=0.155124983896014… is the unique zero in $(1/9,1/6)$ of the polynomial P(d)=1−7d+3d2+4d3−2d4−d5.
Related
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- The Beurling density of the spectrum of self-similar measure generated by Hadamard triple
BibTeX
@misc{liang2026hausdorff,
title = {{On the Hausdorff measure of projections of self-similar sets}},
author = {Chong-Wei Liang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.04684},
eprint = {2609.04684},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.04684},
note = {CC0},
}
RIS
TY - UNPB
TI - On the Hausdorff measure of projections of self-similar sets
AU - Chong-Wei Liang
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.04684
UR - https://arxiv.org/abs/2609.04684
AB - For 0<d≤1/4, let 𝒞(d) be the four-corner Cantor set with Hausdorff dimension s_d=log 4/log(1/d). Peres, Simon, and Solomyak, as well as Mattila, asked for which d the measure Hau^s_d(p_θ(𝒞(d))) is positive for almost every direction θ. It was open for the range 1/9≤ d≤1/6. In this paper, we show that, for δ< d≤1/6, there is a set IP(d) of positive Lebesgue measure such that for almost every θ∈IP(d), Hau^s_d(p_θ(𝒞(d)))=0, where δ=0.155124983896014… is the unique zero in $(1/9,1/6)$ of the polynomial P(d)=1−7d+3d^2+4d^3−2d^4−d^5.
ER -