On multiple Walsh series convergent over cubes

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.