Abstract
There exists a family $\mathcal{B}$ of one-to-one mappings $B \colon \mathbb{Z}\to\mathbb{Z}$ satisfying the condition $B(-n) \equiv -B(n)$ such that for each $B \in \mathcal{B}$ there exists a perfect uniqueness set of positive measure for the $B$-rearranged trigonometric system $\{\exp(iB(n)x)\}$. For a certain wider class of rearrangements of the trigonometric system, the strengthened assertion holds from the Stechkin-Ul'yanov conjecture.
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