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Falconer distance problem in R2\mathbb{R}^2 (C34C_{34})

Falconer distance problem in R2\mathbb{R}^2: determine C34C_{34} or improve its bounds; the repository records 1 (lower) and 54\frac{5}{4} (upper).

MSC 28A75 · Length, area, volume, other geometric measure theoryMSC 28A80 · Fractals

Editorial MSC2020 classification. Original article codes and source evidence are recorded in the literature directory.

Licensed source statementAs of Repository table at commit 2c1968c (2026-09-26); open while the recorded lower and upper bounds differSource & evidence ↓
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Problem statement

The Falconer distance problem threshold C34=sΔ(R2)C_{34} = s_{\Delta}(\mathbb{R}^2) in the plane is defined as

sΔ(R2): := inf⁡{ s∈[0,2] : ∀ compact E⊂R2, dim⁡H(E)>s ⟹ ∣Δ(E)∣>0 }.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⊂R2E\subset \mathbb{R}^2, the distance set is

Δ(E) := { ∣x−y∣ : x,y∈E } ⊂ [0,∞)\Delta(E)\ :=\ \{\,\lvert x-y\rvert\ :\ x,y\in E\,\}\ \subset\ [0,\infty)

[GIOW2018-def-distance-set], dim⁡H\dim_H denotes Hausdorff dimension, and ∣Δ(E)∣\lvert\Delta(E)\rvert denotes the 1-dimensional Lebesgue measure of Δ(E)\Delta(E).

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Background

References (from the repository page)

  • [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⊂RdE \subset \mathbb{R}^d, define the distance set Δ(E)={∣p−p′∣:p,p′∈E}\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\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\dim_H(E) > d+1/2, then L(Δ(E))>0L(\Delta(E)) > 0.”

  • [GIOW2018-wolff-4-3] loc: arXiv v1 PDF p.1, Introduction. quote: “In [37], Wolff proved that if E⊂R2E \subset \mathbb{R}^2 is a compact set with Hausdorff dimension greater than 4/34/3, then Δ(E)\Delta(E) has positive Lebesgue measure.”

  • [GIOW2018-thm-5-4] loc: arXiv v1 PDF p.1, Introduction. quote: “Theorem 1.1. If E⊂R2E \subset \mathbb{R}^2 is a compact set with Hausdorff dimension greater than 5/45/4, then Δ(E)\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≤d/2s \le d/2 there exist sets of Hausdorff dimension ss for which L(Δ(E))=0L(\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.

Progress & source assessment

Repository table (commit 2c1968c, 26 September 2026): best known lower bound 1; best known upper bound 54\frac{5}{4}. Values marked with an asterisk have minimal verification.

Known upper bounds

Bound Reference Comments
3/2=1.53/2=1.5 [Fal1986] Falconer proved (in particular in d=2d=2) that if dim⁡H(E)>3/2\dim_H(E)>3/2 then ∣Δ(E)∣>0\lvert\Delta(E)\rvert>0. [GIOW2018-falconer-3-2]
4/3≈1.33334/3\approx 1.3333 [Wol1999] Wolff improved the planar threshold to dim⁡H(E)>4/3\dim_H(E)>4/3. [GIOW2018-wolff-4-3]
5/4=1.255/4=1.25 [GIOW2018] Guth–Iosevich–Ou–Wang proved that if dim⁡H(E)>5/4\dim_H(E)>5/4 then ∣Δ(E)∣>0\lvert\Delta(E)\rvert>0. [GIOW2018-thm-5-4]

Known lower bounds

Bound Reference Comments
00 Trivial Since dim⁡H(E)≥0\dim_H(E)\ge 0 always, the infimum defining sΔ(R2)s_\Delta(\mathbb{R}^2) is ≥0\ge 0.
11 [Fal1986] Falconer gave examples showing (in general dimension dd) that one cannot expect ∣Δ(E)∣>0\lvert\Delta(E)\rvert>0 below the threshold d/2d/2; in d=2d=2 this yields sΔ(R2)≥1s_\Delta(\mathbb{R}^2)\ge 1. [[GIOW2018-lb-d-2]]
References
  1. [1]Davis, Damek; Ivanisvili, Paata; Tao, Terence (maintainers). Optimization Constants in Mathematics, github.com/teorth/optimizationproblems (commit 2c1968c, 26 September 2026). — Falconer distance problem in $\mathbb{R}^2$ ($C_{34}$) linkApache-2.0 (repository license); constant pages reproduced with attribution; repository references retained.

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