  Schur introduced the problem on the smallest limit point for the arithmetic
means of totally positive conjugate algebraic integers. This area was developed
further by Siegel, Smyth and others. We consider several generalizations of the
problem that include questions on the smallest limit points of symmetric means.
The key tool used in the study is the asymptotic distribution of algebraic
numbers understood via the weak* limits of their counting measures. We
establish interesting properties of the limiting measures, and find the
smallest limit points of symmetric means for totally positive algebraic numbers
of small height.
