Falconer's conjecture, named for Kenneth Falconer, is a problem in geometric measure theory concerning the set of Euclidean distances determined by a compact set of points in d-dimensional space. It states that if such a set has Hausdorff dimension strictly greater than d over 2, then the set of distances between its points must have positive Lebesgue measure.
Facts
StatementIf S is a compact subset of d-dimensional Euclidean space whose Hausdorff dimension is strictly greater than d divided by 2, then the set of Euclidean distances between pairs of points of S has positive Lebesgue measure. 1 Progress Toward ResolutionThe best known general result is due to Erdogan in 2005, who proved the conclusion holds whenever the Hausdorff dimension exceeds d/2 plus 1/3; in the plane, sets of Hausdorff dimension greater than 5/4 are known to satisfy the conjecture; for dimensions greater than 3 the best known threshold, from a 2023 preprint by Du, Ou, Ren and Zhang, is d/2 plus 1/4 minus 1/(8d+4). The conjecture remains open in general. 1 Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Falconer's Conjecture (Wikipedia)
Sources
1. Falconer's Conjecture (Wikipedia)
Statement section
If S is a compact set of points in d-dimensional Euclidean space whose Hausdorff dimension is strictly greater than d/2, then the set of distances between pairs of points in S must have nonzero Lebesgue measure.
Partial results section
compact sets of points whose Hausdorff dimension is greater than d/2 + 1/3 have distance sets with nonzero measure
- Lead section
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