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 ${\displaystyle d}$-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 ${\displaystyle S}$ is a compact set of points in ${\displaystyle d}$-dimensional Euclidean space whose Hausdorff dimension is strictly greater than ${\displaystyle d/2}$, then the conjecture states that the set of distances between pairs of points in ${\displaystyle S}$ must have nonzero Lebesgue measure.^[1]