Math Conjectures

Falconer's Distance Set Conjecture

OPENLandmarkConjectureProposed 1985 · Standard version

Canonical statement

Let d≥2d\geq2 and let E⊂RdE\subset\mathbb R^d be compact. Write dim⁡H\dim_{\mathrm H} for Hausdorff dimension. If
dim⁡H(E)>d2, \dim_{\mathrm H}(E)>\frac d2,
then its distance set Δ(E)={ ∥x−y∥2:x,y∈E }\Delta(E)=\{\,\|x-y\|_2:x,y\in E\,\} has positive one-dimensional Lebesgue measure.
View source LaTeX
Let \(d\geq2\) and let \(E\subset\mathbb R^d\) be compact.
Write \(\dim_{\mathrm H}\) for Hausdorff dimension. If
\[
  \dim_{\mathrm H}(E)>\frac d2,
\]
then its distance set
\(\Delta(E)=\{\,\|x-y\|_2:x,y\in E\,\}\) has positive
one-dimensional Lebesgue measure.
DG-004:[Falconer1985Distance][Mattila1987SphericalAverages][DuOuRenZhang2023Falconer]
Discrete geometryAnalysis and functional analysis

Notes

Falconer's conjecture is a continuous counterpart of the Erdős-type distance problems. Falconer asked in 1985 whether every compact set E⊂RdE\subset\mathbb R^d, d≥2d\ge2, with Hausdorff dimension greater than d/2d/2 must have a distance set Δ(E)\Delta(E) of positive one-dimensional Lebesgue measure; he proved that dimension greater than (d+1)/2(d+1)/2 suffices, while lattice-like examples show the threshold cannot be taken below d/2d/2 [Falconer1985Distance].

Progress has come mainly from Fourier analysis. Mattila recast the problem in terms of spherical averages of Fourier transforms of energy-finite measures, a framework underlying most subsequent work [Mattila1987SphericalAverages]. Modern harmonic-analytic and incidence methods, including the higher-dimensional improvements of Du, Ou, Ren, and Zhang, have pushed the sufficient dimension strictly below the historical thresholds [DuOuRenZhang2023Falconer].

Despite these advances, no proof reaches the conjectured exponent d/2d/2 in every dimension, and the problem remains open; a resolution must close the gap between the best sufficient thresholds currently known and d/2d/2 itself.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.