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.