Point-hyperplane Incidence Geometry and the Log-rank Conjecture

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