On Weyl multipliers of the rearranged trigonometric system

We prove that the condition $\sum_{n=1}^\infty1/(nw(n))<\infty$ is necessary for an increasing sequence of numbers $w(n)$ to be an almost everywhere unconditional convergence Weyl multiplier for the trigonometric system. This property was known long ago for Haar, Walsh, Franklin and some other classical orthogonal systems. The proof of this result is based on a new sharp logarithmic lower bound on $L^2$ for the majorant operator related to the rearranged trigonometric system.
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