Classify intermediate-density line configurations in three dimensions

Characterize the structural features of sets of n lines in R^3 having more than n but at most a constant multiple of n^{3/2} two-rich points.

Background

A line in R3 is determined by four real parameters, so configurations with substantially more than n incidences are overdetermined. The cited theorem gives a classification when the number of two-rich points is much larger than n{3/2}, forcing concentration in planes or degree-2 surfaces.

The intermediate regime remains unresolved: it is highly overdetermined, but the paper reports that no structural information about the line set can currently be proved there.

References

In the regime, $n \ll |P_2(\frak L)| \lesssim n{3/2}$, the problem is still highly overdetermined but we are not able to prove any structural information about $\frak L$.

— Perspectives on the unit distance problem  (2609.10791 - Guth, 9 Sep 2026) in Section 6, subsection “Low degree algebraic structure”