Abstract. It is easy to prove that, given two integers $d\ge2$, $k\ge1$ and a sequence $\omega\in \{0,\ldots,d-1\} ^k$, the frequency of occurrence of $\omega $ in the expansions of the first $N$ integers with respect to the base $d$ tends, when $N\to\infty $, to a limit which only depends on $k$. This paper gives a suitable method to generalize this result to certain algebraic bases, and extends to it the Champernowne's construction.

AMS Subject Classification (1991): 11A63, 11K55, 11M41, 11R04

Received October 28, 1998, and in revised form April 12, 1999. (Registered under 2696/2009.)