The theory of uniform
distribution of sequences of algebraic integers in a fixed algebraic number field K, as
initiated by Kuipers, Niederreiter, and Shiue, is developed from a measure-theoretic
viewpoint. After establishing some general facts in §2, in particular, the
analogy between uniform distribution of sequences of algebraic integers in K
and of sequences of lattice points, a method of enumerating all algebraic
integers in K into a uniformly distributed sequence is discussed in §3. This
enumeration method is useful for the construction of other uniformly distributed
sequences as well and plays a role in the density theory. In §4, a so-called
Banach-Buck measure is defined on the ring of all algebraic integers in K. Various
relations between this measure and the property of uniform distribution are
exhibited. Based on Buck’s general concept of density, the notions of relative
density and of density of sets of algebraic integers in K are introduced in
the final section. Connections among the concepts of uniform distribution,
measurability, and relative density of sequences of algebraic integers in K are
established.
