A typical number is extremely non-normal

Abstract

Fix a positive integer $N\ge 2$. 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 $n\to \infty$. 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.

Updated version includes mention of Olsen and West's work published shortly after first version