The Square Variation of Rearranged Fourier Series

Abstract

We prove that there exists a rearrangement of the first $N$ elements of the trigonometric system such that the $L^2$-norm of the square variation operator is at most $O_{\epsilon}(\log^{9/22+\epsilon}(N))$. This is an improvement over $O(\log^{1/2}(N))$ from the canonical ordering.