A typical number is extremely non-normal

Abstract

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.