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Automatic sequences along squares and primes Drmota, Michael | CIRM

Single angle

V

Research talks

Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they are definitely no "quasi-random" sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal sequences which could be already proved for the Thue-Morse sequence along the subsequence of squares.
This kind of research is very challenging and was mainly motivated by the Gelfond problems for the sum-of-digits function. In particular during the last few years there was a spectacular progress due to the Fourier analytic method by Mauduit and Rivat. In this talk we survey some of these recent developments. In particular we present a new result on subsequences along primes of so-called invertible automatic sequences.
Automatic sequences and their number theoretic properties have been intensively studied during the last 20 or 30 years. Since automatic sequences are quite regular (they just have linear subword complexity) they are definitely no "quasi-random" sequences. However, the situation changes drastically when one uses proper subsequences, for example the subsequence along primes or squares. It is conjectured that the resulting sequences are normal ...

11B85

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Ressources Electroniques

Books & Print journals

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146910683799" class='linkstyle1'>11M06

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Bounded remainder sets for the discrete and continuous irrational rotation Grepstad, Sigrid | CIRM H

Multi angle

y

Research talks

Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$ \sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, Ostrowski and Kesten characterized the intervals with this property.
We will discuss the bounded remainder property for sets in higher dimensions. In particular, we will see that parallelotopes spanned by vectors in $\mathbb{Z}\alpha + \mathbb{Z}^d$ have bounded remainder. Moreover, we show that this condition can be established by exploiting a connection between irrational rotation on $\mathbb{T}^d$ and certain cut-and-project sets. If time allows, we will discuss bounded remainder sets for the continuous irrational rotation $\lbrace t \alpha : t$ $\epsilon$ $\mathbb{R}^+\rbrace$ in two dimensions.
Let $\alpha$ $\epsilon$ $\mathbb{R}^d$ be a vector whose entries $\alpha_1, . . . , \alpha_d$ and $1$ are linearly independent over the rationals. We say that $S \subset \mathbb{T}^d$ is a bounded remainder set for the sequence of irrational rotations $\lbrace n\alpha\rbrace_{n\geqslant1}$ if the discrepancy
$ \sum_{k=1}^{N}1_S (\lbrace k\alpha\rbrace) - N$ $mes(S)$
is bounded in absolute value as $N \to \infty$. In one dimension, Hecke, ...

11K38 ; 11J71 ; 11K06

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Vertex degrees in planar maps Drmota, Michael | CIRM H

Multi angle

y

Research talks

We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even infinitely)-dimensional central limit theorem. We also discuss some possible extension to maps of higher genus.
This is joint work with Gwendal Collet and Lukas Klausner
We consider the family of rooted planar maps $M_\Omega$ where the vertex degrees belong to a (possibly infinite) set of positive integers $\Omega$. Using a classical bijection with mobiles and some refined analytic tools in order to deal with the systems of equations that arise, we recover a universal asymptotic behavior of planar maps. Furthermore we establish that the number of vertices of a given degree satisfies a multi (or even inf...

05A19 ; 05A16 ; 05C10 ; 05C30

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Small sumsets in continuous and discrete settings de Roton, Anne | CIRM H

Multi angle

y

Research talks

Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain how in this problem discrete and continuous setting are linked. This should also illustrate two important principles in additive combinatorics : reduction and rectification.
This talk is partially based on some joint work with Pablo Candela and some other work with Paul Péringuey.
Given a subset A of an additive group, how small can the sumset $A+A = \lbrace a+a' : a, a' \epsilon$ $A \rbrace$ be ? And what can be said about the structure of $A$ when $A + A$ is very close to the smallest possible size ? The aim of this talk is to partially answer these two questions when A is either a subset of $\mathbb{Z}$, $\mathbb{Z}/n\mathbb{Z}$, $\mathbb{R}$ or $\mathbb{T}$ and to explain how in this problem discrete and continuous ...

11B13 ; 11B83 ; 11B75

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Sequences, discrepancies and applications Drmota, Michael ; Tichy, Robert F. | Springer 1997

Ouvrage

V

- 503 p.
ISBN 978-3-540-62606-0

Lecture notes in mathematics , 1651

Localisation : Collection 1er étage

théorie des nombres # discrépance de suites # distribution modulo 1 # irrégularités de distribution # généralisation aléatoire de nombres # intégration numérique # nombre pseudo-aléatoires # distribution uniforme de suite # dispersion # suites dans un espace discret

11Kxx ; 11K06 ; 11K38

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Random Trees : an Interplay between Combinatorics and Probability Drmota, Michael | Springer 2009

Ouvrage

V

- xvii, 458 p.
ISBN 978-3-211-75355-2

Localisation : Ouvrage RdC (DRMO)

arbre aléatoire # graphe aléatoire # profiles

05C05 ; 05C80

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