| # The Falconer distance conjecture | |
| The following describes the scope of the Lean formalization related to the following accompanying paper(s): | |
| - [The Falconer distance conjecture in all dimensions](../../preprints/The-Falconer-distance-conjecture-in-all-dimensions-September-23-2026/paper.pdf) | |
| ## Scope | |
| The formalization proves the Falconer distance conjecture in every dimension. For every integer $d\ge2$ and compact set $E\subset\mathbb R^d$ with Hausdorff dimension greater than $d/2$, the set $\{\lVert x-y\rVert:x,y\in E\}$ has positive Lebesgue measure. The linked statements include both this all-dimensional result and the earlier planar case. | |
| ## Comparator links | |
| | Result | Comparator statement | | |
| | --- | --- | | |
| | Planar Falconer distance theorem | [PlanarFalconer.lean](../ComparatorChallenges/PlanarFalconer.lean) | | |
| | Falconer distance theorem in every dimension | [FalconerAllDimensions.lean](../ComparatorChallenges/FalconerAllDimensions.lean) | | |