[#P12] Falconer distance conjecture
Problem. Let \(d\ge2\) and let \(E\subset\mathbb{R}^d\) be compact with \(\dim_{\mathrm H}(E)>d/2\). Then the distance set \(\{|x-y|:x,y\in E\}\) has positive one-dimensional Lebesgue measure.
1Context
The conjecture is a continuous analogue of discrete distance questions and links fractal geometry with Fourier analysis.
2Problem setup
Definition 1 (The distance set records all Euclidean distances between pairs of points in E). The distance set records all Euclidean distances between pairs of points in E.
Definition 2 (Positive Lebesgue measure). Positive Lebesgue measure is stronger than merely containing infinitely many distances.
Remark 1. The conjecture is a continuous analogue of discrete distance questions and links fractal geometry with Fourier analysis.
3What counts as a solution
- Prove positive measure of the distance set under the stated dimension hypothesis for every d and compact E, or construct a qualifying compact set whose distance set has measure zero.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: The linked 2025 preprint proves Falconer-type conclusions for selected function and set classes. It does not reach the unrestricted Hausdorff-dimension threshold d/2. Exact unresolved remainder: For every dimension d, prove positive measure of the distance set whenever a compact set has Hausdorff dimension greater than d/2, or construct a qualifying counterexample.[1]
1Records
Notes and companion material
Original intake status. The cited 2025 paper proves Falconer's conjecture for selected classes and treats the unrestricted threshold as unresolved. The source and public status were checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- Many improved thresholds and special classes are known. A complete proof must reach the strict d/2 threshold for arbitrary compact sets.
Computational notes
- Finite point clouds approximate particular sets but do not certify Hausdorff dimension or distance-set measure in the limit.
2See also
- Kakeya conjecture in dimensions at least fourgeometric measure theory
- Bochner-Riesz conjecture in higher dimensionsharmonic analysis
- Openness of convolution on l1 of the integersharmonic analysis
How to cite
TheoremDB contributors, “Falconer distance conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/falconer-distance-conjectureThis page as plain text: falconer-distance-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from arxiv.org[1], current as of July 31, 2026.
1References
- Packet source. Minh-Quy Pham, “On Falconer type functions and the distance set problem”. arXiv:2510.15118 (2025). Minh-Quy Pham, arXiv:2510.15118, abstract and applications. ↗preprint · primary source · arXiv:2510.15118, checked 2026-07-31 · checked 2026-07-31Source use: original summary.The cited 2025 paper proves Falconer's conjecture for selected classes and treats the unrestricted threshold as unresolved. The source and public status were checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at abstract and applications to selected Falconer-type classes.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Gives a current special-class result without settling the unrestricted Euclidean threshold.Source named by the research packet.
An original CC0 restatement prepared by TheoremDB maintainers.