Dimension-free bounds and structural results in communication complexity

The purpose of this article is to initiate a systematic study dimension-free relations between basic communication and query complexity measures various matrix norms. In other words, our goal obtain inequalities that bound parameter solely as function another parameter. This in contrast perhaps the more common framework where poly-logarithmic dependencies on number input bits are tolerated. Dimension-free bounds also closely related structural results, one seeks describe structure Boolean matrices functions have low complexity. We prove such theorems for several well operator cases we show do not exist. propose conjectures, establish that, addition applications theory, these problems central characterization idempotents algebra Schur multipliers, could lead new extensions Cohen’s celebrated idempotent theorem regarding Fourier algebra.