On uniform distribution of algebraic numbers

Abstract

Let K be an algebraic number field of degree n + 1 over the rational field R. The conjugates of K are denoted by K(?), , I K(n), it being supposed that K M K(cl are real, K(r?l+), * , K(n) are complex, and moreover that for k =r1+ *, ri +r2 the fields K(k) and K(k+r2) are obtainable from one another by interchanging i( = (1)1/2) and -i (here r1+2r2=n). I assume throughout that r1 ?0 and that n ?1. Numbers in K are represented by small Greek letters, the insertion of superscripts denoting the passage to the corresponding conjugates. Let co, * * , wn be a set of numbers in K which are linearly independent over R; let co = 1; and finally let f(x1, * * *, xn) be a (complex-valued) function of n real variables which is Riemann integrable over the unit cube En: O<Xk<l (k=1, * * *, n). Then, according to theorems of Weyl on uniform distribution,