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Research talks;Dynamical Systems and Ordinary Differential Equations;Number Theory

The Chowla conjecture asserts that the signs of the Liouville function are distributed randomly on the integers. Reinterpreted in the language of ergodic theory this conjecture asserts that the Liouville dynamical system is a Bernoulli system. We prove that ergodicity of the Liouville system implies the Chowla conjecture. Our argument has an ergodic flavor and combines recent results in analytic number theory, finitistic and infinitary decomposition results involving uniformity norms, and equidistribution results on nilmanifolds. The Chowla conjecture asserts that the signs of the Liouville function are distributed randomly on the integers. Reinterpreted in the language of ergodic theory this conjecture asserts that the Liouville dynamical system is a Bernoulli system. We prove that ergodicity of the Liouville system implies the Chowla conjecture. Our argument has an ergodic flavor and combines recent results in analytic number theory, finitistic and infinitary ...

11N60 ; 11B30 ; 11N37 ; 37A45

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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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- x; 476 p.
ISBN 978-2-85629-804-6

Astérisque , 0367;0368

Localisation : Périodique 1er étage;Réserve

topologie # géométrie différentielle # équation aux dérivées partielles # groupe approximatif # analyse fonctionnelle # géométrie algébrique des surfaces K3 # nombre premier # probabilité # preuve formelle

35A05 ; 35L71 ; 37l50 ; 53C20 ; 35B35 ; 35Q20 ; 45K05 ; 60J75 ; 65C05 ; 82C22 ; 82C40 ; 82C80 ; 11B30 ; 03C98 ; 20N99 ; 20F67 ; 57Mxx ; 57M25 ; 57M27 ; 57R17 ; 53C42 ; 49Q20 ; 14J28 ; 14C25 ; 14C20 ; 14C34 ; 14G35 ; 82B44 ; 82B20 ; 60K35 ; 82A70 ; 82B40 ; 11N05 ; 11N13 ; 11N35 ; 11N37 ; 11L05 ; 11T23 ; 03B15 ; 18A15 ; 03B35 ; 68T15 ; 83C05 ; 53C50 ; 53C80 ; 35L72 ; 05C50 ; 15A15 ; 26C10 ; 46L30

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- xiii; 303 p.
ISBN 978-1-4704-2196-0

Graduate studies in mathematics , 0164

Localisation : Collection 1er étage;Réserve

groupe simple fini # groupe de Lie

05C81 ; 11B30 ; 20C33 ; 20D06 ; 20G40

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