Recall that normality is a elementary form of randomness: an infinite word is normal to a given alphabet if all blocks of symbols of the same length occur in the word with the same asymptotic frequency. We consider a notion of independence on pairs of infinite words formalising that two words are independent if no one helps to compress the other using one-to-one finite transducers with two inputs. As expected, the set of independent pairs has Lebesgue measure 1. We prove that not only the join of two normal words is normal, but, more generally, the shuffling with a finite transducer of two normal independent words is also a normal word. The converse of this theorem fails: we construct a normal word as the join of two normal words that are not independent. We construct a word x such that the symbol at position n is equal to the symbol at position 2n. Thus, x is the join of x itself and the subsequence of odd positions of x. We also show that selection by finite automata acting on pairs of independent words preserves normality. This is a counterpart version of Agafonov's theorem for finite automata with two input tapes.
This is joint work with Olivier Carton (Universitéé Paris Diderot) and Pablo Ariel Heiber (Universidad de Buenos Aires).[-]

We fix a positive integer $q\geq 2$. Then every real number $x\in[0,1]$ admits a representation of the form

$x=\sum_{n\geq 1}\frac{a_{n}}{q^{n}}$,

where $a_{n}\in \mathcal{N} :=\{0,1,\ .\ .\ .\ ,\ q-1\}$ for $n\geq 1$. For given $x\in[0,1], N\geq 1$, and $\mathrm{d}=d_{1}\ldots d_{k}\in \mathcal{N}^{k}$ we denote by $\Pi(x,\ \mathrm{d},\ N)$ the frequency of occurrences of the block $\mathrm{d}$ among the first $N$ digits of $x$, i.e.

$\Pi(x, \mathrm{d},N):=\frac{1}{N}|\{0\leq n< N:a_{n+1}=d_{1}, . . . a_{n+k}=d_{k}\}$

from a probabilistic point of view we would expect that in a randomly chosen $x\in[0,1]$ each block $\mathrm{d}$ of $k$ digits occurs with the same frequency $q^{-k}$. In this respect we call a real $x\in[0,1]$ normal to base $q$ if $\Pi(x,\ \mathrm{d},\ N)=q^{-k}$ for each $k\geq 1$ and each $|\mathrm{d}|=k$. When Borel introduced this concept he could show that almost all (with respect to Lebesgue measure) reals are normal in all bases $q\geq 2$ simultaneously. However, still today all constructions of normal numbers have an artificial touch and we do not know whether given reals such as $\sqrt{2},$ log2, $e$ or $\pi$ are normal to a single base.
On the other hand the set of non-normal numbers is large from a topological point of view. We say that a typical element (in the sense of Baire) $x\in[0,1]$ has property $P$ if the set $S :=${$x\in[0,1]:x$ has property $P$} is residual - meaning the countable intersection of dense sets. The set of non-normal numbers is residual.
In the present talk we will consider the construction of sets of normal and non-normal numbers with respect to recent results on absolutely normal and extremely non-normal numbers.[-]