Representation of the diagonal
The following conjecture, if true, would imply the Erdos discrepancy conjecture.
For all C > 0 there exists a diagonal matrix with trace at least C that can be expressed as , where and each Pi and Qi is the characteristic function of a HAP.
Proof of implication
Suppose D is a diagonal matrix with entries bj on the diagonal, with
unbounded, and where and the Pi and Qi are HAPs. Suppose x is a sequence with finite discrepancy C. Then we can write in two ways. On the one hand, , which is unbounded; on the other hand, , a contradiction.
Moses pointed out here that we do not actually need to produce a diagonal matrix D with unbounded trace. It is enough to produce a matrix D such that is unbounded.
Roth's Theorem concerning discrepancy on Arithmetic Progression contains a representation-of-diagonal proof in detail.
This page contains the LaTeX source code for a write-up of a representation-of-diagonal proof that gives the correct bound.