The Square Variation of Rearranged Fourier Series

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 $O_ε(\log^{9/22+ε}(N))$. This is an improvement over $O(\log^{1/2}(N))$ from the canonical ordering.

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