Falconer's distance set conjecture: does $\dim_H E>d/2$ force the distance set of $E\subset\mathbb{R}^d$ to have positive Lebesgue measure?

Falconer's distance set conjecture predicts that for every Borel set $E\subset\mathbb{R}^d$, $d\geq 2$, if the Hausdorff dimension of $E$ exceeds $d/2$, then the distance set $D(E)=\{|x-y|: x,y\in E\}$ has positive Lebesgue measure. Falconer (1985) proved the positive-measure conclusion only under the stronger hypothesis $\dim_H E>(d+1)/2$, and the conjectured threshold $d/2$ has remained out of reach ever since. Bourgain (1994) initiated the threshold-lowering program: within Mattila's Fourier framework it used the fine theory of Fourier restriction to spheres to give the first improvements, in dimensions $2$ and $3$ (planar threshold $13/9$). Work following the same program subsequently reached $4/3$ and then $5/4$ in the plane, $9/5$ in dimension $3$, and $d/2+1/4-1/(8d+4)$ in all dimensions $d\geq 3$, while adjacent results (dimension-type conclusions, extra Fourier hypotheses, finite-field analogues) leave the original measure-theoretic conjecture untouched. As of April 2026 the conjecture is open in every ambient dimension; the residual gap to $d/2$, never reduced below $1/4$ before the 2023 work of Du–Ou–Ren–Zhang, is tied to the discretized sum-product problem. The problem asks for a proof of the conjecture or a counterexample: a Borel set of Hausdorff dimension greater than $d/2$ whose distance set has Lebesgue measure zero.

Background

For an integer $d\geq 2$ and a set $E\subset\mathbb{R}^d$, the distance set of $E$ is $D(E)=\{|x-y|: x,y\in E\}\subset[0,\infty),$ the set of all pairwise Euclidean distances realized inside $E$. The Hausdorff dimension $\dim_H E$ is the standard fractal refinement of dimension, a real number in $[0,d]$ that measures how efficiently $E$ can be covered by small balls; it agrees with the usual dimension on smooth sets but can take non-integer values on fractals. Falconer (1985) proved that if $E$ is a Souslin (that is, analytic) subset of $\mathbb{R}^d$ with $\dim_H E>(d+1)/2$, then $D(E)$ has positive Lebesgue measure, and he conjectured that the optimal threshold is $d/2$: every $E$ with $\dim_H E>d/2$ should have $|D(E)|>0$. This is the Falconer distance set conjecture, the driving open problem of the distance-set program in geometric measure theory.

Mattila (1987) introduced the Fourier-analytic framework in which the conjecture has been attacked ever since. A Frostman measure on $E$ is a probability measure $\mu$ supported on $E$ with $\mu(B(x,r))\leq Cr^s$ for some exponent $s$; such measures exist for every $s<\dim_H E$, so the dimension hypothesis can be traded for the existence of a measure. One then pushes $\mu\times\mu$ forward under the map $(x,y)\mapsto|x-y|$ to obtain the distance measure $\delta(\mu)$, a measure on $[0,\infty)$; if $\delta(\mu)$ has a square-integrable density (that is, $\delta(\mu)\in L^2$), then $D(E)$ has positive Lebesgue measure. By Plancherel's theorem this $L^2$ property is equivalent to quantitative decay, as the frequency radius $r$ grows, of spherical averages of the Fourier transform $\hat\mu$, namely averages of $|\hat\mu(r\omega)|^2$ over directions $\omega$ on the unit sphere $S^{d-1}$. The conjecture thereby becomes a sharp instance of the Fourier restriction/extension problem for spheres, which studies how norms of a function supported on a curved surface control, and are controlled by, norms of its Fourier transform.

Bourgain (1994) obtained the first improvement over Falconer's threshold, in dimensions $d=2$ and $d=3$, by bringing the fine theory of Fourier restriction to spheres to bear on Mattila's framework; in the plane the threshold became $13/9$, and the paper records the conjectured optimal threshold $\dim_H E>d/2$ as the eventual target for all $d\geq 2$. The lineage since: Wolff (1999) reached $4/3$ in the plane via decay of circular means of Fourier transforms of measures; Erdoğan (2005) proved $d/2+1/3$ for $d\geq 3$ via a bilinear Fourier extension theorem; Guth–Iosevich–Ou–Wang (2020) reached $5/4$ in the plane; Du–Guth–Ou–Wang–Wilson–Zhang (2021) reached $9/5$ in $d=3$; Du–Zhang (2019), through sharp $L^2$ estimates for the Schrödinger maximal function (bounds on pointwise convergence of solutions of the Schrödinger equation averaged in time), obtained $d/2+1/4+1/(8d-4)$ in all $d\geq 3$, and Du–Iosevich–Ou–Wang–Zhang (2021) improved even dimensions $d\geq 4$ to $d/2+1/4$; and Du–Ou–Ren–Zhang (2023, revised 2024) proved $d/2+1/4-1/(8d+4)$ for all $d\geq 3$, in the stronger pinned form (some $x\in E$ has $|\{|x-y|: y\in E\}|>0$). The modern proofs combine $L^2$-decoupling (simultaneous control of the Fourier-localized pieces of a function carried on thin curved "plates") with tube-decomposition and radial-projection arguments.

Two families of adjacent results must be distinguished from the conjecture itself. Dimension-type results lower the conclusion: Keleti–Shmerkin (2019) proved lower bounds on $\dim_H D(E)$ for planar sets, and Shmerkin–Wang (2025) proved the dimension version of the conjecture at exactly $d/2$ for sets of equal Hausdorff and packing dimension (packing dimension being a slightly larger, box-counting-flavored notion), obtaining for instance $\dim_H D(E)\geq(\sqrt{5}-1)/2$ for planar sets of dimension $1$; neither gives positive Lebesgue measure. Hypothesis-strengthening results add Fourier-analytic assumptions: Fraser–Pham (2026) beat the $d/2$ threshold for $d\geq 5$ under assumptions on the Fourier dimension and Fourier spectrum of the set (that is, on the decay of Fourier transforms of measures supported on it). The original conjecture — positive Lebesgue measure of $D(E)$ from $\dim_H E>d/2$ alone — remains open in every ambient dimension $d\geq 2$. The obstruction is quantitative: no method had reduced the gap between the best thresholds and $d/2$ below $1/4$ until Du–Ou–Ren–Zhang brought it to $1/4-1/(8d+4)$ for $d\geq 3$, and closing it is expected to require explicit bounds in the discretized sum-product problem (combinatorial growth lower bounds for the sum set $A+A$ and product set $A\cdot A$ of finite point sets), the machinery by which Katz–Tao (2001) and Bourgain (2003) ruled out the hypothetical half-dimensional ring that would make $1/2$ a natural barrier for the distance set at the boundary $\dim_H E=d/2$.

Problem Statement

Prove or refute the Falconer distance set conjecture: for every integer $d\geq 2$ and every Borel set $E\subset\mathbb{R}^d$,

$\dim_H E>\frac{d}{2}\quad\Longrightarrow\quad|D(E)|>0,$

where $D(E)=\{|x-y|: x,y\in E\}$ is the distance set of $E$ (the set of all pairwise Euclidean distances realized inside $E$), $\dim_H$ denotes Hausdorff dimension (the standard fractal dimension, a real number in $[0,d]$ measuring how efficiently the set can be covered by small balls), and $|\cdot|$ denotes Lebesgue measure. It suffices to state the conjecture for compact sets, since every Borel set of Hausdorff dimension $s$ contains compact subsets of dimension arbitrarily close to $s$. A refutation must exhibit, for at least one $d\geq 2$, a Borel set $E\subset\mathbb{R}^d$ with $\dim_H E>d/2$ and $|D(E)|=0$.

The current best thresholds are $5/4$ in the plane (Guth–Iosevich–Ou–Wang 2020) and $d/2+1/4-1/(8d+4)$ for $d\geq 3$ (Du–Ou–Ren–Zhang 2023); closing the remaining gap to $d/2$ in any ambient dimension, under the Hausdorff-dimension hypothesis alone, is the open target. As of April 2026 the conjecture is open in all dimensions.

References

1. K. J. Falconer, "On the Hausdorff dimensions of distance sets", Mathematika 32 (1985), no. 2, 206-212. DOI: 10.1112/s0025579300010998, https://doi.org/10.1112/s0025579300010998
2. Pertti Mattila, "Spherical averages of Fourier transforms of measures with finite energy; dimensions of intersections and distance sets", Mathematika 34 (1987), no. 2, 207-228. DOI: 10.1112/s0025579300013462, https://doi.org/10.1112/s0025579300013462
3. Jean Bourgain, "Hausdorff dimension and distance sets", Israel Journal of Mathematics 87 (1994), no. 1-3, 193-201. DOI: 10.1007/BF02772994, https://doi.org/10.1007/BF02772994
4. Thomas Wolff, "Decay of circular means of Fourier transforms of measures", International Mathematics Research Notices 1999 (1999), no. 10, 547-567. No DOI is registered for the original article.
5. Thomas Wolff, "Addendum to 'Decay of circular means of Fourier transforms of measures'", Journal d'Analyse Mathématique 88 (2002), no. 1, 35-39. DOI: 10.1007/BF02786573, https://doi.org/10.1007/BF02786573
6. Nets Hawk Katz and Terence Tao, "Some Connections between Falconer's Distance Set Conjecture and Sets of Furstenburg Type", New York Journal of Mathematics 7 (2001), 149-187. https://nyjm.albany.edu/j/2001/7-10.html
7. Jean Bourgain, "On the Erdős-Volkmann and Katz-Tao ring conjectures", Geometric and Functional Analysis 13 (2003), no. 2, 334-365. DOI: 10.1007/s000390300008, https://doi.org/10.1007/s000390300008
8. M. B. Erdoğan, "A bilinear Fourier extension theorem and applications to the distance set problem", International Mathematics Research Notices 2005, no. 23, 1411-1425. DOI: 10.1155/IMRN.2005.1411, https://doi.org/10.1155/IMRN.2005.1411
9. Larry Guth, Alex Iosevich, Yumeng Ou, and Hong Wang, "On Falconer's distance set problem in the plane", Inventiones mathematicae 219 (2020), no. 3, 779-830. DOI: 10.1007/s00222-019-00917-x, arXiv:1808.09346, https://arxiv.org/abs/1808.09346
10. Xiumin Du and Ruixiang Zhang, "Sharp $L^2$ estimates of the Schrödinger maximal function in higher dimensions", Annals of Mathematics 189 (2019), no. 3, 837-861. DOI: 10.4007/annals.2019.189.3.4, arXiv:1805.02775, https://arxiv.org/abs/1805.02775
11. Xiumin Du, Larry Guth, Yumeng Ou, Hong Wang, Bobby Wilson, and Ruixiang Zhang, "Weighted restriction estimates and application to Falconer distance set problem", American Journal of Mathematics 143 (2021), no. 1, 175-211. DOI: 10.1353/ajm.2021.0005, arXiv:1802.10186, https://arxiv.org/abs/1802.10186
12. 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), no. 3-4, 1215-1231. DOI: 10.1007/s00208-021-02170-1, https://doi.org/10.1007/s00208-021-02170-1
13. Xiumin Du, Yumeng Ou, Kevin Ren, and Ruixiang Zhang, "New improvement to Falconer distance set problem in higher dimensions", arXiv:2309.04103 (v2, October 2024). https://arxiv.org/abs/2309.04103
14. Tamás Keleti and Pablo Shmerkin, "New Bounds on the Dimensions of Planar Distance Sets", Geometric and Functional Analysis 29 (2019), no. 6, 1886-1948. DOI: 10.1007/s00039-019-00500-9, https://doi.org/10.1007/s00039-019-00500-9
15. Pablo Shmerkin and Hong Wang, "On the Distance Sets Spanned by Sets of Dimension $d/2$ in $\mathbb{R}^{d}$", Geometric and Functional Analysis 35 (2025), no. 1, 283-358. DOI: 10.1007/s00039-024-00696-5, https://doi.org/10.1007/s00039-024-00696-5
16. Jonathan M. Fraser and Thang Pham, "On Fourier decay and the distance set problem", arXiv:2604.19486 (April 2026). https://arxiv.org/abs/2604.19486