Edges of the Quillen-Lichtenbaum range and birational geometry

The Quillen-Lichtenbaum conjecture, now proved by Rost and Voevodsky, in particular asserts that the algebraic K-theory of a smooth C-scheme is equivalent to its topological K-theory in a range. In joint work with Nick Addington, we turned this number into a birational invariant which has the advantage that it is defined for dg-categories and is thus a Morita invariant. I will explain some examples and computations of this birational invariant and its connections with the so-called "cofiber of tau" in motivic homotopy theory.