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Documents  Hennecart, François | enregistrements trouvés : 9

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Research talks;Combinatorics;Number Theory

Erdös and Sárközy asked the maximum size of a subset of the first $N$ integers with no two elements adding up to a perfect square. In this talk we prove that the tight answer is $\frac{11}{32}N$ for sufficiently large $N$. We are going to prove some stability results also. This is joint work with Simao Herdade and Ayman Khalfallah.

05A18 ; 11B75

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Outreach;Mathematics Education and Popularization of Mathematics

Endre Szemerédi (born August 21, 1940) is a Hungarian-American mathematician, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science at Rutgers University since 1986. Szemerédi has won prizes in mathematics and science, including the Abel Prize in 2012. He has made a number of discoveries in combinatorics and computer science, including Szemerédi's theorem, the Szemerédi regularity lemma, the Erdös-Szemeredi theorem, the Hajnal-Szemerédi theorem and the Szemerédi-Trotter theorem. Endre Szemerédi (born August 21, 1940) is a Hungarian-American mathematician, working in the field of combinatorics and theoretical computer science. He has been the State of New Jersey Professor of computer science at Rutgers University since 1986. Szemerédi has won prizes in mathematics and science, including the Abel Prize in 2012. He has made a number of discoveries in combinatorics and computer science, including Szemerédi's theorem, the ...

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Research talks;Combinatorics;Number Theory

This will be a review of structural theory of set addition including recent developments on the polynomial Freiman-Ruzsa conjecture.

11P70 ; 11B13

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Research talks;Combinatorics;Number Theory

Let $k > 2$ be a real number. We inquire into the following question : what is the maximal size (inner Lebesque measure) and the form of a set avoiding solutions to the linear equation $x + y = kz$ ? This problem was used for $k$ an integer larger than 4 to guess the density and the form of a corresponding maximal set of positive integers less than $N$. Nevertheless, in the case $k = 3$, the discrete and the continuous setting happen to be very different. Although the structure of maximal sets in the continuous setting is quite easy to describe for $k$ far enough from 2, it is more difficult to handle as $k$ comes closer to 2. Joint work with Alain Plagne. Let $k > 2$ be a real number. We inquire into the following question : what is the maximal size (inner Lebesque measure) and the form of a set avoiding solutions to the linear equation $x + y = kz$ ? This problem was used for $k$ an integer larger than 4 to guess the density and the form of a corresponding maximal set of positive integers less than $N$. Nevertheless, in the case $k = 3$, the discrete and the continuous setting happen to be very ...

05D05 ; 11Pxx

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Research talks;Combinatorics;Number Theory

We improve a result of Solymosi on sum-products in $\mathbb{R}$, namely, we prove that max $(|A+A|,|AA|\gg |A|^{4/3+c}$, where $c>0$ is an absolute constant. New lower bounds for sums of sets with small product set are found. Previous results are improved effectively for sets $A\subset \mathbb{R}$ with $|AA| \le |A|^{4/3}$. Joint work with I. D. Schkredov.

11B13 ; 11B30 ; 11B75

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Research talks;Combinatorics;Number Theory

Let $H$ be a Krull monoid with finite class group $G$ and suppose that each class contains a prime divisor (rings of integers in algebraic number fields share this property). For each element $a \in H$, its set of lengths $\mathsf L(a)$ consists of all $k \in \mathbb{N} _0$ such that $a$ can be written as a product of $k$ irreducible elements. Sets of lengths of $H$ are finite nonempty subsets of the positive integers, and we consider the system $\mathcal L (H) = \{ \mathsf L (a) \mid a \in H \}$ of all sets of lengths. It is classical that H is factorial if and only if $|G| = 1$, and that $|G| \le 2$ if and only if $|L| = 1$ for each $L \in \mathcal L(H)$ (Carlitz, 1960).

Suppose that $|G| \ge 3$. Then there is an $a \in H$ with $|\mathsf L (a)|>1$, the $m$-fold sumset $\mathsf L(a) + \ldots +\mathsf L(a)$ is contained in $\mathsf L(a^m)$, and hence $|\mathsf L(a^m)| > m$ for every $m \in \mathbb{N}$. The monoid $\mathcal B (G)$ of zero-sum sequences over $G$ is again a Krull monoid of the above type. It is easy to see that $\mathcal L (H) = \mathcal L \big(\mathcal B (G) \big)$, and it is usual to set $\mathcal L (G) := \mathcal L \big( \mathcal B (G) \big)$. In particular, the system of sets of lengths of $H$ depends only on $G$, and it can be studied with methods from additive combinatorics.
The present talk is devoted to the inverse problem whether or not the class group $G$ is determined by the system of sets of lengths. In more technical terms, let $G'$ be a finite abelian group with $|G'| \ge 4$ and $\mathcal L(G) = \mathcal L(G')$. Does it follow that $G$ and $G'$ are isomorphic ?
The answer is positive for groups $G$ having rank at most two $[1]$ and for groups of the form $G = C_{n}^{r}$ with $r \le (n+2)/6$ $[2]$. The proof is based on the characterization of minimal zero-sum sequences of maximal length over groups of rank two, and on the set $\triangle^*(G)$ of minimal distances of $G$ (the latter has been studied by Hamidoune, Plagne, Schmid, and others ; see the talk by Q. Zhong).
Let $H$ be a Krull monoid with finite class group $G$ and suppose that each class contains a prime divisor (rings of integers in algebraic number fields share this property). For each element $a \in H$, its set of lengths $\mathsf L(a)$ consists of all $k \in \mathbb{N} _0$ such that $a$ can be written as a product of $k$ irreducible elements. Sets of lengths of $H$ are finite nonempty subsets of the positive integers, and we consider the system ...

11B30 ; 11R27 ; 13A05 ; 13F05 ; 20M13

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Research talks;Combinatorics;Number Theory

Various problems in additive combinatorics can be translated to a question about incidences in Cartesian products. A well known example is Elekes' treatment of the sum-product problem but there are many more applications of incidence bounds to arithmetic problems. I will review the classical applications and show some recent results.

11B75 ; 11B13 ; 52C10 ; 05Dxx

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- xix; 371 p.
ISBN 978-2-8178-0280-0

Collection IRIS

Localisation : Ouvrage RdC (BRET)

informatique # combinatoires # théorie des graphes # Algorithmes de graphe # arbre # connectivité # graphe plan # théorie algébrique des graphes # colloriage # factorisation # automorphisme # théorie de Ramsey

68-01 ; 05-01 ; 68R10 ; 05C85

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- 45 p.

Localisation : Ouvrage RdC (HENN)

base # méthode du cercle # ordre # théorie additive

11Pxx ; 11B13

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