Hyperplane Furstenberg Sets in Higher Dimensions
Abstract
The hyperplane Furstenberg problem asks how small the Hausdorff dimension of E⊂ℝn can be when a t-dimensional family of affine (n−1)-dimensional hyperplanes meets E in sets of dimension at least s. For 0<s≤ n−1 and 0<t≤ n, we prove the sharp bound in ℝn dimHE≥F(s,t;n,n−1), where F is Gan's Furstenberg index proposed in prime fields. We also construct examples attaining this bound for all parameters. Our main reduction theorem, Theorem 3.1, states that for n≥3, 0<s≤ n−2, and 1<t≤ n, every weighted (s,t) configuration (E,H) at scale δ produces a dual weighted (s,t−1) configuration (P,𝒯) inside a hyperplane Hy0≅ℝn−1. The reduction uses radial projection and energy estimates for Frostman measures. Iterating this reduction and applying the Csörnyei--Stull incidence theorem yields the bound.
Related
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- Kakeya and Furstenberg problems for sticky sets of tubes
- Spread-out Furstenberg sets II: general configurations and fractal restricted projections
- The local dynamical structure of Δ* sets via a new Furstenberg family algebra
- Spread-out Furstenberg sets I: the sticky case
BibTeX
@misc{nie2026hyperplane,
title = {{Hyperplane Furstenberg Sets in Higher Dimensions}},
author = {Peiqi Nie},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.03504},
eprint = {2611.03504},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.03504},
}
RIS
TY - UNPB
TI - Hyperplane Furstenberg Sets in Higher Dimensions
AU - Peiqi Nie
PY - 2026
DA - 2026/11/02
PB - arXiv
DO - 10.48550/arxiv.2611.03504
UR - https://arxiv.org/abs/2611.03504
AB - The hyperplane Furstenberg problem asks how small the Hausdorff dimension of E⊂ℝ^n can be when a t-dimensional family of affine (n−1)-dimensional hyperplanes meets E in sets of dimension at least s. For 0<s≤ n−1 and 0<t≤ n, we prove the sharp bound in ℝ^n dim_HE≥F(s,t;n,n−1), where F is Gan's Furstenberg index proposed in prime fields. We also construct examples attaining this bound for all parameters. Our main reduction theorem, Theorem 3.1, states that for n≥3, 0<s≤ n−2, and 1<t≤ n, every weighted (s,t) configuration (E,H) at scale δ produces a dual weighted (s,t−1) configuration (P,𝒯) inside a hyperplane H_y_0≅ℝ^n−1. The reduction uses radial projection and energy estimates for Frostman measures. Iterating this reduction and applying the Csörnyei--Stull incidence theorem yields the bound.
ER -