[#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.

Point set with measured pairwise distances.

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1Context

The conjecture is a continuous analogue of discrete distance questions and links fractal geometry with Fourier analysis.

2Remarks

Remark 1. The distance set records all Euclidean distances between pairs of points in E.

Remark 2. Positive Lebesgue measure is stronger than merely containing infinitely many distances.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

1Packet records

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Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Falconer distance conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
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.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 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.^[1] Relevance to this problem For Falconer distance conjecture, record falconer-distance-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: 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. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. 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. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
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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
A Davies projection example provable in ZFCgeometric measure theory
Flattest 32-term Littlewood polynomial on the unit circleharmonic analysis

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Plain text

“Falconer distance conjecture.” TheoremDB. P12. Problem statement; statement text SHA-256 aff2c6951d69f5e52cf183f704b01b2e985f1773fe5fa6b580f8d66e7c40c073. https://theoremdb.org/statement/?ref=P12

BibTeX

@misc{theoremdb-problem-aff2c6951d69f5e52cf183f704b01b2e985f1773fe5fa6b580f8d66e7c40c073,
  title = {{Falconer distance conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 aff2c6951d69f5e52cf183f704b01b2e985f1773fe5fa6b580f8d66e7c40c073},
  url = {https://theoremdb.org/statement/?ref=P12}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from arxiv.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P12] Falconer distance conjecture

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Computational notes

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Recent contributions

Computational notes

2See also

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Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance