# P12: Falconer distance conjecture

- ID: `P12`
- Reference: `falconer-distance-conjecture`
- Page: https://theoremdb.org/statements/P12
- Record maturity: Reviewed problem with recorded work

## 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.

### Context

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

### Problem setup

- **Definition (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 (Positive Lebesgue measure).** Positive Lebesgue measure is stronger than merely containing infinitely many distances.
- **Remark.** The conjecture is a continuous analogue of discrete distance questions and links fractal geometry with Fourier analysis.

### What 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.

## Status

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](#reference-1)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- 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 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.
- The formulation and status were checked against the cited paper on 2026-07-22.
- Many improved thresholds and special classes are known. A complete proof must reach the strict d/2 threshold for arbitrary compact sets.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

- Finite point clouds approximate particular sets but do not certify Hausdorff dimension or distance-set measure in the limit.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `falconer-distance-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://arxiv.org/abs/2510.15118
   - Also cited at abstract and applications to selected Falconer-type classes
   - Also cited at Editorial research route recorded 2026-07-31
   - preprint; primary source; arXiv:2510.15118, checked 2026-07-31; checked 2026-07-31
   - Source 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.
   - 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.