Title: Asymptotically Optimal Multi-Paving,
Abstract: Anderson's paving conjecture, now known to hold due to the resolution of the
Kadison-Singer problem asserts that every zero diagonal Hermitian matrix admits
non-trivial pavings with dimension independent bounds. In this paper, we
develop a technique extending the arguments of Marcus, Spielman and Srivastava
in their solution of the Kadison-Singer problem to show the existence of
non-trivial pavings for collections of matrices. We show that given zero
diagonal Hermitian contractions $A^{(1)}, \cdots, A^{(k)} \in M_n(\mathbb{C})$
and $\epsilon > 0$, one may find a paving $X_1 \amalg \cdots \amalg X_r = [n]$
where $r \leq 18k\epsilon^{-2}$ such that, \[\lambda_{max} (P_{X_i} A^{(j)}
P_{X_i}) < \epsilon, \quad i \in [r], \, j \in [k].\] As a consequence, we get
the correct asymptotic estimates for paving general zero diagonal matrices;
zero diagonal contractions can be $(O(\epsilon^{-2}),\epsilon)$ paved. As an
application, we give a simplified proof wth slightly better estimates of a
theorem of Johnson, Ozawa and Schechtman concerning commutator representations
of zero trace matrices.