  We investigate the square variation operator $V^2$ (which majorizes the
partial sum maximal operator) on general orthonormal systems (ONS) of size $N$.
We prove that the $L^2$ norm of the $V^2$ operator is bounded by $O(\ln(N))$ on
any ONS. This result is sharp and refines the classical Rademacher-Menshov
theorem. We show that this can be improved to $O(\sqrt{\ln(N)})$ for the
trigonometric system, which is also sharp. We show that for any choice of
coefficients, this truncation of the trigonometric system can be rearranged so
that the $L^2$ norm of the associated $V^2$ operator is $O(\sqrt{\ln\ln(N)})$.
We also show that for $p>2$, a bounded ONS of size $N$ can be rearranged so
that the $L^2$ norm of the $V^p$ operator is at most $O_p(\ln \ln (N))$
uniformly for all choices of coefficients. This refines Bourgain's work on
Garsia's conjecture, which is equivalent to the $V^{\infty}$ case. Several
other results on operators of this form are also obtained. The proofs rely on
combinatorial and probabilistic methods.

Classical Analysis and ODEs;Probability