Uniform distribution in solvable groups

Abstract

We study the relations of Hartmann and unitary uniform distribution in solvable groups, in particular in semidirect products of Abelian groups. In every nilpotent group these notions of uniform distribution coincide, but, in general, they are different in solvable groups, as is demonstrated by the motion group of the plane. However, we show that Hartmann and unitary uniform distribution coincide in every solvable analytic group whose Lie algebra has no purely imaginary roots. Finally, we give two six-dimensional solvable analytic groups with the same set of roots, such that the concepts of uniform distribution coincide in one group and differ in the other.