The Bourgain-Tzafriri conjecture and concrete constructions of non-pavable projections

It is known that the Kadison-Singer Problem (KS) and the Paving Conjecture (PC) are equivalent to the Bourgain-Tzafriri Conjecture (BT). Also, it is known that (PC) fails for $2$-paving projections with constant diagonal $1/2$. But the proofs of this fact are existence proofs. We will use variations of the discrete Fourier Transform matrices to construct concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense. In 1989, Bourgain and Tzafriri showed that the class of zero diagonal matrices with small entries (on the order of $\le 1/log^{1+\epsilon}n$, for an $n$-dimensional Hilbert space) are {\em pavable}. It has always been assumed that this result also holds for the BT-Conjecture - although no one formally checked it. We will show that this is not the case. We will show that if the BT-Conjecture is true for vectors with small coefficients (on the order of $\le C/\sqrt{n}$) then the BT-Conjecture is true and hence KS and PC are true

Phase retrieval by hyperplanes

We show that a scalable frame does phase retrieval if and only if the hyperplanes of its orthogonal complements do phase retrieval. We then show this result fails in general by giving an example of a frame for $\mathbb R^3$ which does phase retrieval but its induced hyperplanes fail phase retrieval. Moreover, we show that such frames always exist in $\mathbb R^d$ for any dimension $d$. We also give an example of a frame in $\mathbb R^3$ which fails phase retrieval but its perps do phase retrieval. We will also see that a family of hyperplanes doing phase retrieval in $\mathbb R^d$ must contain at least $2d-2$ hyperplanes. Finally, we provide an example of six hyperplanes in $\mathbb R^4$ which do phase retrieval