Geodesic
Sets of absolute convergence of double trigonometric series
Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 473-480.

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We obtain a sufficient condition for a set of plane measure zero to be a set of absolute convergence (an A.C.-set) for a double trigonometric series. Specifically, let $y=f(x)$ ($0\leqslant x\leqslant2\pi$) be a smooth curve and let $\bigvee\limits_0^{2\pi}\ln|f'(x)|\infty$. Then, any set of positive linear measure lying on this curve is an A.C.-set. 
@article{MZM_1972_11_5_a0,
     author = {R. A. Avetisyan},
     title = {Sets of absolute convergence of double trigonometric series},
     journal = {Matemati\v{c}eskie zametki},
     pages = {473--480},
     publisher = {mathdoc},
     volume = {11},
     number = {5},
     year = {1972},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a0/}
}
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R. A. Avetisyan. Sets of absolute convergence of double trigonometric series. Matematičeskie zametki, Tome 11 (1972) no. 5, pp. 473-480. http://geodesic.mathdoc.fr/item/MZM_1972_11_5_a0/