Falconer's Conjecture

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a compact set $E\subseteq\mathbb R^d$, its Euclidean distance set is \[ \Delta(E)=\{\|x-y\|:x,y\in E\}\subseteq[0,\infty). \] Write $\mathcal L^1$ for one-dimensional Lebesgue measure. The Hausdorff dimension $\dim_H E$ is $\inf\{s\ge0:\mathcal H^s(E)=0\}$, where \[ \mathcal H^s(E)= \lim_{\delta\downarrow0}\inf \left\{\sum_i(\operatorname{diam}U_i)^s: E\subseteq\bigcup_iU_i,\ \operatorname{diam}U_i<\delta\right\}. \] An inessential positive normalization factor in $\mathcal H^s$ does not change this dimension.

Statement recorded in the supplied source.

For every integer $d\ge2$ and every compact set $E\subseteq\mathbb R^d$, does \[ \dim_H E>\frac d2 \quad\Longrightarrow\quad \mathcal L^1(\Delta(E))>0 \] hold? The same implication in dimension one is also included by the literal dimension-free wording, but the higher-dimensional assertion above is the standard Falconer problem. [S2] [S3]

Short English statement

Does every compact set of Hausdorff dimension greater than half the ambient dimension determine a positive-measure set of distances?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1164 (GEO-009). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Larry Guth, Alex Iosevich, Yumeng Ou and Hong Wang, On Falconer's distance set problem in the plane, Inventiones Mathematicae 219 (2020), 779--830, Section 1; arXiv:1808.09346. https://arxiv.org/abs/1808.09346.
• S3 Xiumin Du, Alex Iosevich, Yumeng Ou, Hong Wang and Ruixiang Zhang, An improved result for Falconer's distance set problem in even dimensions, Mathematische Annalen 380 (2021), 1215--1231, Section 1; arXiv:2006.06833. https://arxiv.org/abs/2006.06833.