Determine whether near-extremal point-line configurations have arithmetic structure

Determine whether sufficiently dense point-line incidence configurations in the real plane necessarily possess an arithmetic structure, such as arising from integer points or points over a number field after a projective transformation.

Background

The best known constructions for point-line incidences with close to the Szemerédi–Trotter exponent are based on integer points, polynomial rings, or number fields. However, the incidence hypothesis itself is purely real-algebraic and does not mention arithmetic objects.

The paper presents the existence of a structure or classification theorem for such configurations as a major open question. It also emphasizes that no precise general conjecture had previously been formulated, motivating the later dimension and degree conjectures.

References

Proving some kind of structure theorem for the point-line incidence problem in the plane is a major open question in incidence geometry.

— Perspectives on the unit distance problem  (2609.10791 - Guth, 9 Sep 2026) in Section 6, subsection “Arithmetic structure?” and concluding paragraphs

It is not yet clear whether integer points and number fields play a central role in incidence problems or whether these are the only examples we have found because of lack of imagination.

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