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H 1 Normal and non-normal numbers

Auteurs : Madritsch, Manfred (Auteur de la Conférence)
CIRM (Editeur )

Résumé : 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.

Keywords : normal numbers; radix representation; symbolic dynamics

Codes MSC :
11A63 - Radix representation; digital problems
11K16 - Normal numbers, radix expansions, Pisot numbers, Salem numbers, good lattice points, etc.

 Informations sur la Vidéo Langue : Anglais Date de publication : 22/02/2021 Date de captation : 04/02/2021 Collection : Research talks Format : MP4 Durée : 00:59:52 Domaine : Probability & Statistics ; Number Theory Audience : Chercheurs ; Doctorants , Post - Doctorants Download : https://videos.cirm-math.fr/2021-02-04_madritsch.mp4 Informations sur la rencontre Nom de la rencontre : Jean-Morlet Chair 2020 (2) - Workshop: Discrepancy Theory and Applications - Part 2 / Chaire Jean-Morlet 2020 (2) - Workshop : Théorie de la discrépance et applications - Part 2Organisateurs de la rencontre : Madritsch, Manfred ; Rivat, Joël ; Tichy, RobertDates : 04/02/2021 - 05/02/2021 Année de la rencontre : 2021 URL Congrès : https://www.chairejeanmorlet.com/2641.htmlCitation Data DOI : 10.24350/CIRM.V.19714903 Cite this video as: Madritsch, Manfred (2021). Normal and non-normal numbers. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19714903 URI : http://dx.doi.org/10.24350/CIRM.V.19714903

### Voir aussi

Bibliographie

1. BOREL, M. Émile. Les probabilités dénombrables et leurs applications arithmétiques. Rendiconti del Circolo Matematico di Palermo (1884-1940), 1909, vol. 27, no 1, p. 247-271. - https://doi.org/10.1007/BF03019651

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