Furstenberg's problem about $\times 2,\times 3$-invariant measures generalizes naturally to automorphisms of higher-dimensional tori. While in the one-dimensional case Rudolph's theorem is the best answer under the additional assumption of positive entropy, the analogue in the higher-dimensional setting was for a long time a partial result by Katok and Spatzier; for some (TNS) actions the full generalization of Rudolph's theorem was known, for others additional ergodicity assumptions were necessary. I will present a joint work with Elon Lindenstrauss where we prove a complete generalization of Rudolph's theorem.