Vol. 61, No. 1, 1975

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Banach-Buck measure, density, and uniform distribution in rings of algebraic integers

Siu Kwong Lo and Harald G. Niederreiter

Vol. 61 (1975), No. 1, 191–208

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.

Mathematical Subject Classification 2000
Primary: 10K05, 10K05
Secondary: 65C10
Received: 6 September 1974
Published: 1 November 1975
Siu Kwong Lo
Harald G. Niederreiter
Department of Mathematics
National University of Singapore
2 Science Drive 2