On the optimal paving over MASAs in von Neumann algebras

We prove that if $A$ is a singular MASA in a II$_1$ factor $M$ and $\omega$ is a free ultrafilter, then for any $x\in M\ominus A$, with $\|x\|\leq 1$, and any $n\geq 2$, there exists a partition of $1$ with projections $p_1, p_2, ..., p_n\in A^\omega$ (i.e. a {\it paving}) such that $\|\Sigma_{i=1}^n p_i x p_i\|\leq 2\sqrt{n-1}/n$, and give examples where this is sharp. Some open problems on optimal pavings are discussed.