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# Documents  11A63 | enregistrements trouvés : 12

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## On ellipsephic integers Dartyge, Cécile | CIRM H

Post-edited

Number Theory

The term " ellipsephic " was proposed by Christian Mauduit to denote the integers with missing digits in a given basis. This talk is a survey on several results on the multiplicative properties of these integers.

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## Normal and non-normal numbers Madritsch, Manfred | CIRM H

Multi angle

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.
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. ...

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## Prime numbers with preassigned digits Swaenepoel, Cathy | CIRM H

Multi angle

Number Theory

Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain’s strategy, is based on the circle method and combines techniques from harmonic analysis together with results on zeros of Dirichlet $L$-functions, notably a very sharp zero-free region due to Iwaniec. Bourgain (2015) estimated the number of prime numbers with a proportion $c$ > 0 of preassigned digits in base 2 ($c$ is an absolute constant not specified). We present a generalization of this result in any base $g$ ≥ 2 and we provide explicit admissible values for the proportion $c$ depending on $g$. Our proof, which adapts, develops and refines Bourgain’s strategy, is based on the circle method and combines techniques from harmonic analysis ...

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## On the digits of primes and squares Rivat, Joël | CIRM H

Multi angle

Number Theory

I will give a survey of our results on the digits of primes and squares (joint works with Michael Drmota and Christian Mauduit).

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## Yet another characterization of the Pisot conjecture Mercat, Paul | CIRM H

Multi angle

In the way of Arnoux-Ito, we give a general geometric criterion for a subshift to be measurably conjugated to a domain exchange and to a translation on a torus. For a subshift coming from an unit Pisot irreducible substitution, we will see that it becomes a simple topological criterion. More precisely, we define a topology on $\mathbb{Z}^d$ for which the subshift has pure discrete spectrum if and only if there exists a domain of the domain exchange on the discrete line that has non-empty interior. We will see how we can compute exactly such interior using regular languages. This gives a way to decide the Pisot conjecture for any example of unit Pisot irreducible substitution.
Joint work with Shigeki Akiyama.
In the way of Arnoux-Ito, we give a general geometric criterion for a subshift to be measurably conjugated to a domain exchange and to a translation on a torus. For a subshift coming from an unit Pisot irreducible substitution, we will see that it becomes a simple topological criterion. More precisely, we define a topology on $\mathbb{Z}^d$ for which the subshift has pure discrete spectrum if and only if there exists a domain of the domain ...

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## Algebraic sums and products of univoque bases Dajani, Karma | CIRM H

Multi angle

Given $x\in(0, 1]$, let ${\mathcal U}(x)$ be the set of bases $\beta\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^{\infty}{{d_i}/{\beta^i}}$. In 2014, Lü, Tan and Wu proved that ${\mathcal U}(x)$ is a Lebesgue null set of full Hausdorff dimension. In this talk, we will show that the algebraic sum ${\mathcal U}(x)+\lambda {\mathcal U}(x)$, and the product ${\mathcal U}(x)\cdot {\mathcal U}(x)^{\lambda}$ contain an interval for all $x\in (0, 1]$ and $\lambda\ne 0$. As an application we show that the same phenomenon occurs for the set of non-matching parameters associated with the family of symmetric binary expansions studied recently by the first speaker and C. Kalle.
This is joint work with V. Komornik, D. Kong and W. Li.
Given $x\in(0, 1]$, let ${\mathcal U}(x)$ be the set of bases $\beta\in(1,2]$ for which there exists a unique sequence $(d_i)$ of zeros and ones such that $x=\sum_{i=1}^{\infty}{{d_i}/{\beta^i}}$. In 2014, Lü, Tan and Wu proved that ${\mathcal U}(x)$ is a Lebesgue null set of full Hausdorff dimension. In this talk, we will show that the algebraic sum ${\mathcal U}(x)+\lambda {\mathcal U}(x)$, and the product ${\mathcal U}(x)\cdot {\mathcal ... Déposez votre fichier ici pour le déplacer vers cet enregistrement. ## Primes with missing digits Maynard, James | CIRM H Multi angle Number Theory We will talk about recent work showing there are infinitely many primes with no$7$in their decimal expansion. (And similarly with$7$replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most$X^{1-c}$elements less than$X$) which is typically vey difficult. The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, combinatorial geometry as well as tools from analytic number theory. We will talk about recent work showing there are infinitely many primes with no$7$in their decimal expansion. (And similarly with$7$replaced by any other digit.) This shows the existence of primes in a 'thin' set of numbers (sets which contain at most$X^{1-c}$elements less than$X\$) which is typically vey difficult.
The proof relies on a fun mixture of tools including Fourier analysis, Markov chains, Diophantine approximation, com...

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## Four faces of number theory Bringmann, Kathrin ; Bugeaud, Yann ; Hilberdink, Titus ; Sander, Jürgen | European Mathematical Society 2015

Ouvrage

- viii; 190 p.
ISBN 978-3-03719-142-2

EMS series of lectures in mathematics

Localisation : Ouvrage RdC (FOUR)

théorie des nombres # transcendance # nombre algébrique # théorème subspatial de Schmidt # fraction continue # expansion digitale # approximation Diophantienne

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## The magic numbers of the professor O'Shea, Owen ; Underwood, Dudley | The Mathematical Association of America 2007

Ouvrage

- 168 p.
ISBN 978-0-88385-557-7

Localisation : Loisir bastide

récréation mathématique # problème numérique # curiosités numérique # symbolisme des nombres

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## Substitutions in dynamics, arithmetics and combinatorics Pytheas Fogg, N. ; Berthé, V. ; Ferenczi, S. ; Mauduit, Christian ; Siegel, A. | Springer 2002

Ouvrage

- 402 p.
ISBN 978-3-540-44141-0

Lecture notes in mathematics , 1794

Localisation : Collection 1er étage

combinatoire des mots # transcendance # dynamique symbolique # fractal # théorie ergodique # suite de Sturm # fraction continue # système dynamique # suite automatique

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## Nouveau traité de mathématiques générales. Tome II :analyse, mécanique, théorie des erreurs Fabry, Eugène | Librairie Scientifique J. Hermann 1925

Ouvrage

- 276 p.

Localisation : Réserve

analyse # cinématique # courbure # dynamique des systèmes # dérivée partielle # enseignement # intégrale # mathématique générale # mécanique # résistance des matériaux # statique # série trigonométrique # théorie des erreurs # vecteur # équation différentielle

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## Fractals arithmétiques Long, Lun-Hai | I.R.M.A.;U.L.P. 1993

Thèse

- 113 p.

Localisation : Salle de manutention

dimension de Hausdorff # ensemble auto-affine # ensemble à similitude interne # fractals # matrice de structure # mesure de Hausdorff # système de numération # système de restes

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