In a recent post, I discussed the disproof of Erdős’ unit distance conjecture. That problem asks how many pairs of points can be separated by one prescribed distance. A closely related problem asks the opposite kind of question: how few different distances can occur among all pairs of points?
On August 14, 2026, Jonathan Tidor, Hung-Hsun Hans Yu, and Dmitrii Zakharov posted a 147-page preprint resolving this problem in three-dimensional space, up to a subpolynomial factor (Tidor et al. 2026). Their result determines the correct exponent and shows that the three-dimensional integer lattice is essentially optimal.
Let be a finite set of points in . Its distance set is Thus is the number of distinct positive distances determined by pairs of points of . Define The Erdős distinct distances problem asks for the asymptotic growth of as tends to infinity.
The natural example in is a cubic section of the integer lattice. Suppose first that and let The square of the distance between two points of has the form It is therefore an integer between and . Consequently, For an arbitrary , one can take points from a cube of side length , giving
In his original 1946 paper, Erdős conjectured that lattice constructions of this kind are asymptotically optimal (Erdős 1946). In the plane, the integer lattice determines approximately distances. The celebrated theorem of Guth and Katz states that every set of points in the plane determines at least a constant multiple of distinct distances (Guth and Katz 2015). Thus the planar problem is solved up to a factor of order .
The higher-dimensional problem proved much more resistant. Before the work of Tidor, Yu, and Zakharov, the best lower bound in followed by combining the Guth–Katz theorem with a bootstrapping argument of Solymosi and Vu (Solymosi and Vu 2008). It gave There was therefore a substantial gap between the exponent in the best lower bound and the exponent supplied by the integer lattice.
The new theorem closes this gap.
Since tends to zero, the theorem and the lattice construction together give Equivalently, the lower bound can be written in the form The factor multiplying is subpolynomial: it is smaller than a fixed power of , but larger than the reciprocal of every fixed power of . Thus the exponent is now known to be optimal. The result does not yet prove the stronger estimate with an absolute implied constant.
The contrast with the unit distance problem is striking. For unit distances, the long-standing expectation that a lattice should be extremal was recently disproved. For distinct distances in , the lattice heuristic survives: the exponent supplied by the cubic lattice is now known to be the correct one.
Two natural questions remain. The first is whether the subpolynomial loss can be eliminated, giving The second is what happens in dimensions . The -dimensional integer lattice gives but the corresponding lower bound is still unknown. Some components of the new proof may extend to higher dimensions, but geometric triality is a feature particular to three-dimensional rigid motions. Thus the resolution of the three-dimensional problem is both the end of a long-standing question and the beginning of a new set of incidence-geometric problems.