Point-hyperplane incidence geometry and the log-rank conjecture

We study the log-rank conjecture from the perspective of incidence geometry and present a reformulation as well as a strengthening. The reformulation involves point sets in $\mathbb{R}^d$ that are covered in many ways by constant sized sets of parallel hyperplanes. We show that the log-rank conjecture is equivalent to the implication that all such configurations contain a subspace that accounts for a large fraction of the incidences, in the sense of containing a large fraction of the points and being contained in a large fraction of the hyperplanes. In other words the log-rank conjecture is equivalent to asserting that the point-hyperplane incidence graph for such configurations has a large complete bipartite subgraph. The strengthening of the log-rank conjecture comes from relaxing the requirements that the set of hyperplanes be parallel. Motivated by the connections above we revisit some well-studied questions in point-hyperplane incidence geometry and present some improvements. We give a simple probabilistic argument for the existence of complete bipartite subgraphs of density $\Omega(\epsilon^{2d}/d)$ in any $d$-dimensional configuration with incidence density $\epsilon$, matching previously known results qualitatively. We also improve an upper-bound construction of Apfelbaum and Sharir, yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is $\Omega(1/\sqrt d)$.