Asymptotically Optimal Multi-Paving

Abstract

Abstract Anderson’s paving conjecture, now known to be true [14], asserts that every zero-diagonal matrix admits a nontrivial paving with dimension independent bounds. We study this problem for a collection of matrices and show that given $k$ zero-diagonal $n\times n$ Hermitian matrices $A_1,\ldots ,A_k$ and $\epsilon>0$ there are diagonal projections $P_1,\ldots ,P_r$ with $\sum _{j\le r} P_j=I$ such that $||P_{j}A_iP_{j}||\le \epsilon ||A_i||$ for $i\le k, j\le r$, that is, a simultaneous paving of the matrices, with $r\le 18k/\epsilon ^2$. As a consequence, we get the optimal asymptotic estimates for paving a single zero-diagonal (not necessarily Hermitian) matrix: every square zero-diagonal complex matrix can be $\epsilon -$paved using $O(\epsilon ^{-2})$ blocks, improving the previously known bound of $O(\epsilon ^{-8})$. We use our result to strengthen a result of Johnson–Ozawa–Schechtman on commutator representations of zero trace matrices.