Higher-dimensional incidence independence theory

Develop a theory of independence and dependence for incidences among points, lines, and higher-dimensional affine subspaces in higher-dimensional vector spaces.

Background

The paper studies rank-2 incidence geometries consisting of points and lines in the projective plane. It proposes extending the rigidity-matrix and self-stress viewpoint to incidence structures involving subspaces of higher dimension.

References

Instead of just using points and lines, we could use also higher dimensional affine subspaces of some vector space and ask about independence and dependence of incidences there.

— Counting for rigidity under projective transformations in the plane  (2503.07228 - Berman et al., 10 Mar 2025) in Section "Open problems and future work," subsection "Extensions to higher dimensions"

We do not believe that \cref{thm:incidence-bound-codim-1} is tight, even without the logarithmic loss. In the situation that is relevant to the distinct distances problem, we conjecture the following bound

— The Erdős distinct distances problem in $\mathbb{R}^3$  (2608.14454 - Tidor et al., 14 Aug 2026) in Conjecture labeled \cref{conj:2-flat--3-flat}, Section 10.2, Weak bounds on very rich partial symmetries