Geodesic
On the Cantor--Lebesgue theorem for double trigonometric series
Matematičeskie zametki, Tome 12 (1972) no. 1, pp. 13-17.

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Suppose that on some measurable set $E\subset\mathbf{T}^2$, $\mu(E)>2/3$, $$ A_\nu(x)=\sum_{n_1^2+n_2^2=\nu}c_{n_1,n_2}e^{2\pi i(n_1x_1+n_2x_2)}\to0\qquad(\nu\to\infty). $$ Then $$ \sum_{n_1^2+n_2^2=\nu}|c_{n_1,n_2}|^2\to0\qquad(\nu\to\infty). $$ 
@article{MZM_1972_12_1_a1,
     author = {S. B. Stechkin},
     title = {On the {Cantor--Lebesgue} theorem for double trigonometric series},
     journal = {Matemati\v{c}eskie zametki},
     pages = {13--17},
     publisher = {mathdoc},
     volume = {12},
     number = {1},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_12_1_a1/}
}
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S. B. Stechkin. On the Cantor--Lebesgue theorem for double trigonometric series. Matematičeskie zametki, Tome 12 (1972) no. 1, pp. 13-17. http://geodesic.mathdoc.fr/item/MZM_1972_12_1_a1/