Favard Length and Lipschitz Graph Structure

Definitions and mathematical statement

Let $\mathcal H^1$ denote one-dimensional Hausdorff measure and let $S^1=\{\theta\in{\mathbb{R}}^2:|\theta|=1\}$. For $\theta\in S^1$, let $\pi_\theta:{\mathbb{R}}^2\to{\mathbb{R}}\theta$ be orthogonal projection. Under the usual convention, a nonempty set $E\subset[0,1]^2$ of positive diameter is $1$-Ahlfors regular if there exists $A\geq1$ such that \[ A^{-1}r\leq\mathcal H^1(E\cap B(x,r))\leq Ar \qquad(x\in E,\ 0<r\leq\operatorname{diam}E), \] where $B(x,r)$ is the Euclidean open ball. The source does not specify the regularity constant $A$, its permitted range, or an explicit measurability convention. The measure and integral expressions below are understood to be well defined; the source does not explicitly impose compactness or other hypotheses ensuring this.

Define the Favard length by \[ \operatorname{Fav}(E)=\int_{S^1}\mathcal H^1(\pi_\theta(E))\,{\,\mathrm{d}}\theta. \] Here ${\,\mathrm{d}}\theta$ may conventionally be taken as arc-length measure on $S^1$; the supplied record does not specify whether it is normalized. Any quantitative formulation must use one convention consistently.

A Lipschitz graph in an arbitrary direction means a set \[ \Gamma=\{x_0+te+f(t)n:t\in{\mathbb{R}}\}, \] where $x_0\in{\mathbb{R}}^2$, $(e,n)$ is an orthonormal basis of ${\mathbb{R}}^2$, and $f:{\mathbb{R}}\to{\mathbb{R}}$ satisfies \[ \operatorname{Lip}(f)=\sup_{s\ne t}\frac{|f(s)-f(t)|}{|s-t|}<\infty. \] The notation $\operatorname{Lip}(\Gamma)$ in the question is interpreted as the Lipschitz constant in such a representation. The source does not specify a preferred graph direction.

Quantitative question. On the literal reading of the source's $C$-dependent bounds, is it true that for every $C>0$ there exist numbers $L(C)<\infty$ and $m(C)>0$ such that every $1$-Ahlfors regular set $E\subset[0,1]^2$ satisfying $\operatorname{Fav}(E)\geq C$ admits a Lipschitz graph $\Gamma$ with \[ \operatorname{Lip}(\Gamma)\leq L(C), \qquad \mathcal H^1(E\cap\Gamma)\geq m(C)? \] Unresolved dependence. The supplied notation is $\operatorname{Lip}(\Gamma)\lesssim_C1$ and $\mathcal H^1(E\cap\Gamma)\gtrsim_C1$, but it is not stated whether the Ahlfors regularity constant is fixed implicitly. A formulation for sets with a prescribed regularity bound $A$, allowing $L=L(C,A)$ and $m=m(C,A)$, is a distinct quantitative formulation. The record does not resolve which dependence is intended.

Short English statement

Does a positive lower bound on the average length of the orthogonal projections of a $1$-Ahlfors regular planar set force a quantitatively large part of that set to lie on a Lipschitz graph with quantitatively bounded slope? The intended dependence on the Ahlfors regularity constant remains unspecified. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, data/unsolvedmath/problems.json, permanent ID $30005544$, title: Favard Length and Lipschitz Graph Structure; exact source problem code: catalogue entry. The supplied record attributes the problem to the 2023 Oberwolfach workshop report and names no individual proposer.
• Incidence Problems in Harmonic Analysis, Geometric Measure Theory, and Ergodic Theory (2023), Oberwolfach Reports, pp. 1397--1452, DOI: https://doi.org/10.4171/owr/2023/25. Supplied report link, labeled OWR 2023 25: https://publications.mfo.de/bitstream/handle/mfo/4055/OWR_2023_25.pdf?sequence=1. No precise page or conjecture number within the report is supplied.
• Supplied literature reference labeled Dąbrowski: https://doi.org/10.1112/plms.70037. The record supplies no article title or precise theorem identifying the claimed special case for irregular direction sets.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem partially solved, describes the full Favard-length-to-Lipschitz-graph statement as conjectural, and attributes a substantial special case for irregular direction sets to Dąbrowski. This is supplied dated curation, not an independently verified literature assessment.
• The supplied statement status is corrected_verified, with a supplied assertion that the clean statement was checked against the official report. That verification assertion has not been independently checked here. The retained original-statement fragment reads: "Can this result be made quantitative? The following conjecture is wide open." Its antecedent result is not included in the supplied record; this historical wording is distinct from the dated partially-solved label.