A Continuum Beck-type Theorem for Hyperplanes

TL;DR

This work proves a sharp continuum Beck-type theorem for hyperplanes in $\mathbb{R}^n$, establishing that for a Borel set $X$ in which no union of affine flats of total dimension at most $n-1 concentrates $X$, the set of hyperplanes spanned by $X$ has dimension at least $n\min\{\dim X,1\}$. The authors develop an inductive framework based on decomposing $X$ into irreducible $s$-Frostman pieces on flats, organizing these pieces in $c$-stable position, and reducing to a minimal configuration of flats. Core tools include thin $k$-planes graphs, radial projection results in projection theory (via Osw–Shmerkin–Wang and Ren), and a key lemma that transfers thin-planes from subcollections to the full minimal setup. The combination yields a quantitative continuum Beck-type bound for hyperplanes, with sharpness demonstrated via NC (non-concentration) constraints, and provides a robust structural approach that generalizes prior planar line results to higher-dimensional hyperplanes. The results have implications for understanding incidence geometry and projection phenomena in fractal geometry.

Abstract

We prove a sharp continuum Beck-type theorem for hyperplanes. Our work is inspired by foundational work of Beck on the discrete problem, as well as refinements due to Do and Lund. The inductive proof uses recent breakthrough results in projection theory by Orponen--Shmerkin--Wang and Ren, who proved continuum Beck-type theorems for lines in $\mathbb{R}^2$ and $\mathbb{R}^n$.

Theorems & Definitions (98)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3: Theorem 1.1, orponen2022kaufman
• Theorem 1.4: Theorem 1.1, ren2023discretized
• Definition 1.5
• Lemma 1.6
• Lemma 1.7
• Definition 1.8
• Lemma 2.1
• proof : Proof of Lemma \ref{['lem:decomposition']}