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Hausdorff Dimension of Weighted Singular Vectors
Abstract
Let d≥2 and let w=(w1,…,wd) satisfy w1≥⋯≥ wd>0 and ∑i wi=1. Set s*=d−(1+w1)−1. We prove that there exist constants Cd,w>0 and ε0=ε0(d,w)>0 such that for all 0<ε<ε0, dimHDIw(ε)≤ s*+Cd,w√ε. Together with the lower bound of Kim--Park, this gives the exact formula dimHSing(w)=s*.
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BibTeX
@misc{yang2026hausdorff,
title = {{Hausdorff Dimension of Weighted Singular Vectors}},
author = {Bohan Yang and Tianru Zhu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.35009},
eprint = {2609.35009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.35009},
}
RIS
TY - UNPB
TI - Hausdorff Dimension of Weighted Singular Vectors
AU - Bohan Yang
AU - Tianru Zhu
PY - 2026
DA - 2026/09/28
PB - arXiv
DO - 10.48550/arxiv.2609.35009
UR - https://arxiv.org/abs/2609.35009
AB - Let d≥2 and let w=(w_1,…,w_d) satisfy w_1≥⋯≥ w_d>0 and ∑_i w_i=1. Set s_*=d−(1+w_1)^−1. We prove that there exist constants C_d,w>0 and ε_0=ε_0(d,w)>0 such that for all 0<ε<ε_0, dim_HDI_w(ε)≤ s_*+C_d,w√ε. Together with the lower bound of Kim--Park, this gives the exact formula dim_HSing(w)=s_*.
ER -