Falconer distance problem in $\mathbf{R}^2$
Description of constant
The Falconer distance problem threshold $C_{34} = s_{\Delta}(\mathbb{R}^2)$ in the plane is defined as \(s_{\Delta}(\mathbb{R}^2) :\ :=\ \inf\Bigl\{\, s\in[0,2]\ :\ \forall\ \text{compact }E\subset\mathbb{R}^2,\ \dim_H(E)>s\ \Longrightarrow\ \lvert\Delta(E)\rvert>0 \,\Bigr\}.\) where for a compact set $E\subset \mathbb{R}^2$, the distance set is \(\Delta(E)\ :=\ \{\,\lvert x-y\rvert\ :\ x,y\in E\,\}\ \subset\ [0,\infty)\) [GIOW2018-def-distance-set], $\dim_H$ denotes Hausdorff dimension, and $\lvert\Delta(E)\rvert$ denote the 1-dimensional Lebesgue measure of $\Delta(E)$.
Known upper bounds
| Bound | Reference | Comments |
|---|---|---|
| $3/2=1.5$ | [Fal1986] | Falconer proved (in particular in $d=2$) that if $\dim_H(E)>3/2$ then $\lvert\Delta(E)\rvert>0$. [GIOW2018-falconer-3-2] |
| $4/3\approx 1.3333$ | [Wol1999] | Wolff improved the planar threshold to $\dim_H(E)>4/3$. [GIOW2018-wolff-4-3] |
| $5/4=1.25$ | [GIOW2018] | Guth–Iosevich–Ou–Wang proved that if $\dim_H(E)>5/4$ then $\lvert\Delta(E)\rvert>0$. [GIOW2018-thm-5-4] |
Known lower bounds
| Bound | Reference | Comments |
|---|---|---|
| $0$ | Trivial | Since $\dim_H(E)\ge 0$ always, the infimum defining $s_\Delta(\mathbb{R}^2)$ is $\ge 0$. |
| $1$ | [Fal1986] | Falconer gave examples showing (in general dimension $d$) that one cannot expect $\lvert\Delta(E)\rvert>0$ below the threshold $d/2$; in $d=2$ this yields $s_\Delta(\mathbb{R}^2)\ge 1$. [[GIOW2018-lb-d-2]] |
Additional comments and links
- The Falconer distance conjecture in the plane predicts that the lower bound of $1$ is sharp. [GIOW2018-conj-plane] [GIOW2018-thm-5-4] [GIOW2018-lb-d-2]
References
- [GIOW2018] Guth, Larry; Iosevich, Alex; Ou, Yumeng; Wang, Hong. On Falconer’s distance set problem in the plane. Inventiones mathematicae 219 (3) (2020), 779–830. DOI: 10.1007/s00222-019-00922-7. Google Scholar. arXiv PDF
- [GIOW2018-def-distance-set] loc: arXiv v1 PDF p.1, Introduction. quote: “For a set $E \subset \mathbb{R}^d$, define the distance set $\Delta(E) = {\lvert p-p’\rvert : p, p’ \in E}$.”
- [GIOW2018-conj-plane] loc: arXiv v1 PDF p.1, Introduction. quote: “This led him to conjecture that if $\dim_H(E) > d/2$, then the Lebesgue measure of the distance set is positive. This is known as the Falconer Distance Conjecture.”
- [GIOW2018-falconer-3-2] loc: arXiv v1 PDF p.1, Introduction. quote: “He proved that if $\dim_H(E) > d+1/2$, then $L(\Delta(E)) > 0$.”
- [GIOW2018-wolff-4-3] loc: arXiv v1 PDF p.1, Introduction. quote: “In [37], Wolff proved that if $E \subset \mathbb{R}^2$ is a compact set with Hausdorff dimension greater than $4/3$, then $\Delta(E)$ has positive Lebesgue measure.”
- [GIOW2018-thm-5-4] loc: arXiv v1 PDF p.1, Introduction. quote: “Theorem 1.1. If $E \subset \mathbb{R}^2$ is a compact set with Hausdorff dimension greater than $5/4$, then $\Delta(E)$ has positive Lebesgue measure.”
- [GIOW2018-lb-d-2] loc: arXiv v1 PDF p.1, Introduction. quote: “Using an example based on the integer lattice, he showed for every $s \le d/2$ there exist sets of Hausdorff dimension $s$ for which $L(\Delta(E)) = 0$.”
-
[Fal1986] Falconer, K. J. On the Hausdorff dimensions of distance sets. Mathematika 32 (1985), no. 2, 206–212. DOI: 10.1112/S0025579300010998. Google Scholar.
- [Wol1999] Wolff, Thomas. Decay of circular means of Fourier transforms of measures. International Mathematics Research Notices 1999 (10), 547–567. DOI: 10.1155/S1073792899000288. Google Scholar.
Contribution notes
Prepared with assistance from ChatGPT 5.2 Pro.