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 Difficulty Hint Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Falconer's Conjecture (Wikipedia)
Posed By
Source Kenneth Falconer (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, difficulty hint
an unsolved problem concerning the sets of Euclidean distances between points in compact
In Branch: Measure Theory, Lead section
In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer
View the SourceKenneth Falconer (Wikipedia)
Posed By: Kenneth Falconer, ResearchQuote, Posed By: Kenneth Falconer, Research
He formulated Falconer's conjecture on the dimension of distance sets
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