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Hyperplane Incidences and Distance Sets in Higher Dimensions — arXiv:2609.38742

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.38742
License (as archived): CC0-1.0
Archived: 2026-10-07T14:19:16.565Z
SHA-256 of the archived text: abb6c1ee54e1e2c740de361f29e5beddc22f6a7cdeae2bf5b1277287c0a34b9c
Stable URL of this page: https://metraipolis.com/documento/dc_4db00522b2fc

How to cite

Hyperplane Incidences and Distance Sets in Higher Dimensions — arXiv:2609.38742 — arXiv (en), https://arxiv.org/abs/2609.38742 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_4db00522b2fc (Metraipolis archive dc_4db00522b2fc, sha256 abb6c1ee54e1e2c740de361f29e5beddc22f6a7cdeae2bf5b1277287c0a34b9c)

BibTeX

@misc{dc4db00522b2fc,
  title = {Hyperplane Incidences and Distance Sets in Higher Dimensions — arXiv:2609.38742},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.38742},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_4db00522b2fc; sha256 abb6c1ee54e1e2c740de361f29e5beddc22f6a7cdeae2bf5b1277287c0a34b9c},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.38742. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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 $E\subset \R^d$ is a Borel set of dimension $\dim_H(E) > d/2$, then \begin{equation*} \sup_{x\in E} \dim_H(\Delta_x(E)) \geq 2/3, \end{equation*} where $\Delta_x(E)$ 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 $d\ge 2$, $s \in (d-2, d-1]$ and $t \in (0, d]$.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-10-01 in math.CO).
Categorie arXiv: Classical Analysis and ODEs (math.CA) ; Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.38742 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-07T14:05:16.450Z dalla pagina https://arxiv.org/catchup/math.CO/2026-10-01?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).