  Fix a positive integer $N\geq2$. For a real number $x\in[0,1]$ and a digit
$i\in\{0, 1,...,N-1\}$, let $\Pi_i(x, n)$ denote the frequency of the digit $i$
among the first $n$ $N$-adic digits of $x$. It is well-known that for a typical
(in the sense of Baire) $x\in[0, 1]$, the frequencies diverge as
$n\rightarrow\infty$. In this paper we provide a substantial strengthening of
this result. Namely, we show that for a typical $x\in[0, 1]$ any regular linear
average of the sequence $(\Pi_i(x, n))_n$ also diverges spectacularly.
