Actions by automorphisms of commutative compact groups with countably multiple Lebesgue component in the spectrum (Q1814460)

Let \(G\) be a countable discrete abelian group, \(\widehat G\) be its dual, \(T^*_ g\) be a representation of a finitely generated subgroup of \(G\) by automorphisms of \(\widehat G\), \(T_ g\) be the action induced by \(T^*_ g\) on \(G\). If \(T^*_ g\) is ergodic then \(T_ g\) has a countable set of similar orbits, if moreover \(T^*_ g\) is of Lebesgue character then the spectrum of \(T^*_ g\) has a Lebesgue component of countable multiplicity (theorem 1).