Falconer's conjecture

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 d-dimensional spaces. Intuitively, it states that a set of points that is large in its Hausdorff dimension must determine a set of distances that is large in measure. More precisely, if S is a compact set of points in d-dimensional Euclidean space whose Hausdorff dimension is strictly greater than d/2, then the conjecture states that the set of distances between pairs of points in S must have nonzero Lebesgue measure.

Falconer (1985) proved that Borel sets with Hausdorff dimension greater than (d + 1)/2 have distance sets with nonzero measure. 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 ${\displaystyle (-\epsilon ,\epsilon )}$ for some ${\displaystyle \epsilon >0}$. 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.

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