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# Documents  Dartyge, Cécile | enregistrements trouvés : 15

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## Post-edited  Polignac numbers and the consecutive gaps between primes Pintz, János (Auteur de la Conférence) | CIRM (Editeur )

We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a closer analysis of the methods of Goldston-Pintz-Yildirim, Green-Tao, Zhang and Maynard-Tao, respectively.
We prove a number of surprising results about gaps between consecutive primes and arithmetic progressions in the sequence of generalized twin primes which could not have been proven without the recent new results of Zhang, Maynard and Tao. The presented results are far from being immediate consequences of the results about bounded gaps between primes: they require various new ideas, other important properties of the applied sieve function and a ...

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## Post-edited  Large gaps between primes in subsets Maynard, James (Auteur de la Conférence) | CIRM (Editeur )

All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long strings of consecutive composite values of a polynomial. This is joint work with Ford, Konyagin, Pomerance and Tao.
All previous methods of showing the existence of large gaps between primes have relied on the fact that smooth numbers are unusually sparse. This feature of the argument does not seem to generalise to showing large gaps between primes in subsets, such as values of a polynomial. We will talk about recent work which allows us to show large gaps between primes without relying on smooth number estimates. This then generalizes naturally to show long ...

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## Post-edited  On ellipsephic integers Dartyge, Cécile (Auteur de la Conférence) | CIRM (Editeur )

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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## Multi angle  Angles of Gaussian primes Rudnick, Zeév (Auteur de la Conférence) | CIRM (Editeur )

Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a random matrix model and by function field considerations.
Fermat showed that every prime $p = 1$ mod $4$ is a sum of two squares: $p = a^2 + b^2$, and hence such a prime gives rise to an angle whose tangent is the ratio $b/a$. Hecke showed, in 1919, that these angles are uniformly distributed, and uniform distribution in somewhat short arcs was given in by Kubilius in 1950 and refined since then. I will discuss the statistics of these angles on fine scales and present a conjecture, motivated by a ...

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## Multi angle  Small sumsets in continuous and discrete settings de Roton, Anne (Auteur de la Conférence) | CIRM (Editeur )

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 ...

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## Multi angle  Large character sums Lamzouri, Youness (Auteur de la Conférence) | CIRM (Editeur )

For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
$M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$.
This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we shall present recent results on higher order character sums. In the first part, we discuss even order characters, in which case we obtain optimal omega results for $M(\chi)$, extending and refining Paley's construction. The second part, joint with Alexander Mangerel, will be devoted to the more interesting case of odd order characters, where we build on previous works of Granville and Soundararajan and of Goldmakher to provide further improvements of the Pólya-Vinogradov and Montgomery-Vaughan bounds in this case. In particular, assuming GRH, we are able to determine the order of magnitude of the maximum of $M(\chi)$, when $\chi$ has odd order $g \geq 3$ and conductor $q$, up to a power of $log_4 q$ (where $log_4$ is the fourth iterated logarithm).
For a non-principal Dirichlet character $\chi$ modulo $q$, the classical Pólya-Vinogradov inequality asserts that
$M (\chi) := \underset{x}{max}$$| \sum_{n \leq x}$$\chi(n)| = O (\sqrt{q} log$ $q)$.
This was improved to $\sqrt{q} log$ $log$ $q$ by Montgomery and Vaughan, assuming the Generalized Riemann hypothesis GRH. For quadratic characters, this is known to be optimal, owing to an unconditional omega result due to Paley. In this talk, we ...

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## Multi angle  Bounded remainder sets for the discrete and continuous irrational rotation Grepstad, Sigrid (Auteur de la Conférence) | CIRM (Editeur )

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, ...

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## Multi angle  Symbolic bounded remainder sets Berthé, Valérie (Auteur de la Conférence) | CIRM (Editeur )

Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded discrepancy provides particularly strong convergence properties of ergodic sums. It is also closely related to the notions of balance in word combinatorics.
Discrepancy is a measure of equidistribution for sequences of points. We consider here discrepancy in the setting of symbolic dynamics and we discuss the existence of bounded remainder sets for some families of zero entropy subshifts, from a topological dynamics viewpoint. A bounded remainder set is a set which yields bounded discrepancy, that is, the number of times it is visited differs by the expected time only by a constant. Bounded ...

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## Multi angle  Non-normal sets Queffélec, Martine (Auteur de la Conférence) | CIRM (Editeur )

The talk will have two main parts:
At the beginning of his mathematical career, Christian inherits from his advisor, Gérard Rauzy, some appetite for distributions of arithmetical sequences, with a special interest for those obtained by simple algorithmic constructions. I will discuss his result on “normal sets associated with substitutions" [?] and continue with recent developments and open questions on sets of non-normal numbers, in a more general setting.
We have written 48 joint papers with Christian. As a tribute to his memory I will present a short survey of our most important papers and recall some of the memorable moments of our cooperation.
The talk will have two main parts:
At the beginning of his mathematical career, Christian inherits from his advisor, Gérard Rauzy, some appetite for distributions of arithmetical sequences, with a special interest for those obtained by simple algorithmic constructions. I will discuss his result on “normal sets associated with substitutions" [?] and continue with recent developments and open questions on sets of non-normal numbers, in a more ...

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## Multi angle  Christian Mauduit in ergodic theory Weiss, Benjamin (Auteur de la Conférence) | CIRM (Editeur )

While most of Christian’s work was in number theory he made important contributions to several aspects of ergodic theory throughout his career. I will discuss some of these and their impact on later developments.

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## Multi angle  Possibles et impossibles en mathématiques Banderier, Cyril (Auteur de la Conférence) | CIRM (Editeur )

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## Single angle  Some news on bilinear decomposition of the Möbius function Ramaré, Olivier (Auteur de la Conférence) | CIRM (Editeur )

This talk presents some news on bilinear decompositions of the Möbius function. In particular, we will exhibit a family of such decompositions inherited from Motohashi's proof of the Hoheisel Theorem that leads to
$\sum_{n\leq X,(n,q)=1) }^{} \mu (n)e(na/q)\ll X\sqrt{q}/\varphi (q)$
for $q \leq X^{1/5}$ and any $a$ prime to $q$.

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## Single angle  On the proximity of additive and multiplicative functions de Koninck, Jean-Marie (Auteur de la Conférence) | CIRM (Editeur )

Given an additive function $f$ and a multiplicative function $g$, let
$E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$
We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that
$E(\varepsilon ,g;x)\gg \frac{x}{\left ( \log \log x\right )^{1+\varepsilon }}$,
while we prove that $E(\varepsilon ,g;x)=o(x)$ as $x\rightarrow \infty$ for every multiplicative function $g$.
Given an additive function $f$ and a multiplicative function $g$, let
$E(f,g;x)=\#\left \{ n\leq x:f(n)=g(n) \right \}$
We study the size of $E(f,g;x)$ for those functions $f$ and $g$ such that $f(n)\neq g(n)$ for at least one value of $n> 1$. In particular, when $f(n)=\omega (n)$ , the number of distinct prime factors of $n$ , we show that for any $\varepsilon >0$ , there exists a multiplicative function $g$ such that

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