PreprintCC BY 4.0arXiv
Hyperplane Incidences and Distance Sets in Higher Dimensions
Abstract
We generalize Ren and Wang's incidence bound between points and lines in R2 \cite{RenWan23} to higher dimensions. We show how to use this incidence bound to improve the best known bound for Falconer's distance set problem in R3 and in R4. We show that if d=3 or d=4, and E⊂ Rd is a Borel set of dimension dimH(E) > d/2, then supx∈ E dimH(Δx(E)) ≥ 2/3, where Δx(E) is the pinned distance set of E with respect to x. We also show how the incidence bound can be used to generalize the planar Furstenberg set bound, to sets in Rd that contain a t-dimensional set of hyperplanes, each of which contains an s-dimensional set of points, for any d≥ 2, s ∈ (d−2, d−1] and t ∈ (0, d].
Related
- Hyperplane Furstenberg Sets in Higher Dimensions
- Geometric progressions in distance sets in odd dimensions over prime fields
- Equivalence of freeness and higher order freeness of hyperplane arrangements
- Fractal dimensions and quasisymmetric packing-minimalities of some homogeneous Moran sets
- Conditions for traceability under a bound on the size of even-distance sets
BibTeX
@misc{csornyei2026hyperplane,
title = {{Hyperplane Incidences and Distance Sets in Higher Dimensions}},
author = {Marianna Csornyei and D. M. Stull},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.38742},
eprint = {2609.38742},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.38742},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hyperplane Incidences and Distance Sets in Higher Dimensions
AU - Marianna Csornyei
AU - D. M. Stull
PY - 2026
DA - 2026/09/30
PB - arXiv
DO - 10.48550/arxiv.2609.38742
UR - https://arxiv.org/abs/2609.38742
AB - We generalize Ren and Wang's incidence bound between points and lines in R^2 \cite{RenWan23} to higher dimensions. We show how to use this incidence bound to improve the best known bound for Falconer's distance set problem in R^3 and in R^4. We show that if d=3 or d=4, and E⊂ R^d is a Borel set of dimension dim_H(E) > d/2, then sup_x∈ E dim_H(Δ_x(E)) ≥ 2/3, where Δ_x(E) is the pinned distance set of E with respect to x. We also show how the incidence bound can be used to generalize the planar Furstenberg set bound, to sets in R^d that contain a t-dimensional set of hyperplanes, each of which contains an s-dimensional set of points, for any d≥ 2, s ∈ (d−2, d−1] and t ∈ (0, d].
ER -