Geodesic
On sets of absolute convergence
Sbornik. Mathematics, Tome 38 (1981) no. 2, pp. 245-254

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In this paper the author indicates a certain class of planar curves such that any set of positive linear measure on them is a set of absolute convergence for double trigonometric series. Bibliography: 11 titles. 
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     author = {R. A. Avetisyan},
     title = {On sets of absolute convergence},
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R. A. Avetisyan. On sets of absolute convergence. Sbornik. Mathematics, Tome 38 (1981) no. 2, pp. 245-254. http://geodesic.mathdoc.fr/item/SM_1981_38_2_a5/