Point-hyperplane incidence geometry and the log-rank conjecture

We study the log-rank conjecture from the perspective of point-hyperplane incidence geometry. We formulate the following conjecture: Given a point set in $\mathbb{R}^d$ that is covered by constant-sized sets of parallel hyperplanes, there exists an affine subspace that accounts for a large (i.e., $2^{-{\operatorname{polylog}(d)}}$) fraction of the incidences. Alternatively, our conjecture may be interpreted linear-algebraically as follows: Any rank-$d$ matrix containing at most $O(1)$ distinct entries in each column contains a submatrix of fractional size $2^{-{\operatorname{polylog}(d)}}$, in which each column contains one distinct entry. We prove that our conjecture is equivalent to the log-rank conjecture. Motivated by the connections above, we revisit well-studied questions in point-hyperplane incidence geometry without structural assumptions (i.e., the existence of partitions). We give an elementary argument for the existence of complete bipartite subgraphs of density $\Omega(\epsilon^{2d}/d)$ in any $d$-dimensional configuration with incidence density $\epsilon$. We also improve an upper-bound construction of Apfelbaum and Sharir (SIAM J. Discrete Math. '07), yielding a configuration whose complete bipartite subgraphs are exponentially small and whose incidence density is $\Omega(1/\sqrt d)$. Finally, we discuss various constructions (due to others) which yield configurations with incidence density $\Omega(1)$ and bipartite subgraph density $2^{-\Omega(\sqrt d)}$. Our framework and results may help shed light on the difficulty of improving Lovett's $\tilde{O}(\sqrt{\operatorname{rank}(f)})$ bound (J. ACM '16) for the log-rank conjecture; in particular, any improvement on this bound would imply the first bipartite subgraph size bounds for parallel $3$-partitioned configurations which beat our generic bounds for unstructured configurations.