In geometric measure theory, Falconer's conjecture, named after Kenneth Falconer, is an unsolved problem concerning the sets of Euclidean distances between points in compact
, then the conjecture states that the set of distances between pairs of points in
[1] Falconer (1985) proved that Borel sets with Hausdorff dimension greater than
[2] He motivated this result as a multidimensional generalization of the Steinhaus theorem, a previous result of Hugo Steinhaus proving that every set of real numbers with nonzero measure must have a difference set that contains an interval of the form
[3] It may also be seen as a continuous analogue of the Erdős distinct distances problem, which states that large finite sets of points must have large numbers of distinct distances.
[4] Erdoğan (2005) proved that compact sets of points whose Hausdorff dimension is greater than
have distance sets with nonzero measure; for large values of
[5] For points in the Euclidean plane, Borel sets of Hausdorff dimension greater than 5/4 (or
according to a preprint by Du, Ou, Ren and Zhang[7][8] A variant of Falconer's conjecture states that, for points in the plane, a compact set whose Hausdorff dimension is greater than or equal to one must have a distance set of Hausdorff dimension one.
This follows from the results on measure for sets of Hausdorff dimension greater than 5/4.
[9] Proving a bound strictly greater than 1/2 for the dimension of the distance set in the case of compact planar sets with Hausdorff dimension at least one would be equivalent to resolving several other unsolved conjectures.
These include a conjecture of Paul Erdős on the existence of Borel subrings of the real numbers with fractional Hausdorff dimension, and a variant of the Kakeya set problem on the Hausdorff dimension of sets such that, for every possible direction, there is a line segment whose intersection with the set has high Hausdorff dimension.
For non-Euclidean distance functions in the plane defined by polygonal norms, the analogue of the Falconer conjecture is false: there exist sets of Hausdorff dimension two whose distance sets have measure zero.