Mixed Determinants and the Kadison-Singer problem

Understand

We adapt the arguments of Marcus, Spielman and Srivastava in their proof of the Kadison-Singer problem to prove improved paving estimates.

• Working with Anderson's paving formulation of Kadison-Singer instead of Weaver's vector balancing version, we show that the machinery of interlacing polyomials due to Marcus, Spielman and Srivastava works in this setting as well.
• The relevant expected characteristic polynomials turn out to be related to the so called "mixed determinants" that have been carefully studied by Borcea and Branden.
• This technique allows us to show that any projection with diagonal entries strictly less than $\frac{1}{4}$ can be two paved, matching recent results of Bownik, Casazza, Marcus and Speegle, though our estimates are asymptotically weaker.