Geodesic
Uniqueness sets of positive measure for the trigonometric system
Izvestiya. Mathematics , Tome 86 (2022) no. 6, pp. 1179-1203

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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.
Keywords: trigonometric system, Fourier series, sets of uniqueness, $V$-sets. 
@article{IM2_2022_86_6_a7,
     author = {M. G. Plotnikov},
     title = {Uniqueness sets of positive measure for the trigonometric system},
     journal = {Izvestiya. Mathematics },
     pages = {1179--1203},
     publisher = {mathdoc},
     volume = {86},
     number = {6},
     year = {2022},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2022_86_6_a7/}
}
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M. G. Plotnikov. Uniqueness sets of positive measure for the trigonometric system. Izvestiya. Mathematics , Tome 86 (2022) no. 6, pp. 1179-1203. http://geodesic.mathdoc.fr/item/IM2_2022_86_6_a7/