Geodesic
On multiple Walsh series convergent over cubes
Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 57-73

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We consider Walsh functions on the binary group $G$ and study uniqueness sets for $N$-fold multiple Walsh series under convergence over cubes (in other words, $U_{N,\mathrm{cube}}$-sets). We prove that every finite set is a $U_{N,\mathrm{cube}}$-set, construct examples of countable $U_{N,\mathrm{cube}}$-sets and non-empty perfect $U_{N,\mathrm{cube}}$-sets, and give an example of a $U_{N,\mathrm{cube}}$-set having the maximum possible Hausdorff dimension. 
@article{IM2_2007_71_1_a4,
     author = {M. G. Plotnikov},
     title = {On multiple {Walsh} series convergent over cubes},
     journal = {Izvestiya. Mathematics },
     pages = {57--73},
     publisher = {mathdoc},
     volume = {71},
     number = {1},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a4/}
}
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M. G. Plotnikov. On multiple Walsh series convergent over cubes. Izvestiya. Mathematics , Tome 71 (2007) no. 1, pp. 57-73. http://geodesic.mathdoc.fr/item/IM2_2007_71_1_a4/