 Séminaire tournant de théorie des nombres Grenoble, Lyon et Saint-Etienne (France)

# Journée 23

Programme de la Journée 23
Jeudi 29 mars 2018

Les exposés auront lieu dans la Salle 4 au rez-de-chaussée de l'Institut Fourier, de 10h30 à 16h.

10h    Café devant la Salle 4.

10h30-11h20, Bruno Martin (Université de Calais), Comportement multifractal de la fonction de Brjuno

Effectuer l'analyse multifractale d'une fonction f : R --> R localement intégrable consiste à déterminer la dimension de Hausdorff des ensembles de points au voisinage desquels la régularité en moyenne (en un sens à préciser) de f est la même. Dans un travail récent avec Stéphane Jaffard, nous réalisons l'analyse multifractale de la fonction de Brjuno : cette fonction, dont la définition fait intervenir le développement en fraction continue d'un nombre réel, joue un rôle fondamental dans la théorie des systèmes dynamiques engendrés par les itérations d'une fonction holomorphe au voisinage d'un point fixe.

11h40-12h30, Tamara Servi (Université Paris 7), Interactions between functional transcendence, resolution of singularities and model theory

Given two (real or complex) analytic functions f and g, we aim to understand under which hypotheses f and g are locally interdefinable, in the context of o-minimal structures. Local interdefinability generalises (differential) algebraic dependence: locally interdefinable functions satisfy relations expressible by some first-order formulas, which are in general more complicated than differential algebraic relations.
There are situations where algebraic independence implies independence with respect to local definability: the real exponential function and the sine function, which satisfy the conclusion of Ax’s functional transcendence theorem, are not locally interdefinable [Bianconi]. The same holds for complex exponentiation and any Weierstrass P-function [Jones, Kirby, Servi]. Two Weierstrass P-functions are locally interdefinable if and only if one can be obtained from the other by isogeny and Schwarz reflection  [Jones, Kirby, Servi]. There are complex analytic functions which are locally interdefinable and which cannot be obtained from one another by elementary operations such as Schwarz reflection, composition, derivation and extracting implicit functions [Jones, Kirby, Le Gal, Servi]. In the case of real analytic functions, it is possible to give an analytic characterisation of all the functions g which are locally definable from f [Le Gal, Servi, Vieillard-Baron].
I will discuss the aforementioned results and their proofs, which rely on an interaction between methods from functional transcendence, resolution of singularities and model theory.

12h30    Buffet en salle de lecture au 2ème étage de l'Institut Fourier.

14h00-14h50, Clemens Müllner (Université Lyon 1), Normal subsequences of automatic sequences

In 2013 Drmota, Mauduit and Rivat observed that the subsequence along the squares t(n^2) of the Thue-Morse sequence t(n) (that can be defined by t(n) = s_2(n) mod 2, where s_2(n) denotes the binary sum-of-digits function) is a normal sequence on the alphabet {0,1}. This means that any block (b_1,...,b_k) in {0,1}^k appears with the same frequency 1/2^k.
The purpose of this talk is to discuss this result also from a more general point of view.  The Thue-Morse sequence is a special case of an k-automatic sequence A(n), that is, a sequence where the n-th element is the output of a deterministic finite state automaton, where the input is the base k expansion of n. Automatic sequence have a sub-linear subword complexity - so they are far from being normal. Furthermore, linear subsequences of automatic sequences are again automatic sequence and, therefore, have again sub-linear subword complexity. However, when we consider a subsequence A(φ(n)), where φ(n)/n tend to infinity the situation can change completely - as t(n^2) shows. Thus, we discuss the question for which subsequences - and for which automatic sequences - we may expect a normal sequence.
Recent results in this direction, e.g., by Lukas Spiegelhofer and the author (who considered φ(n) = [n^c] for the Thue-Morse sequence, where 1 < c < 3/2) and  by the author (who considered block-additive functions like the Rudin-Shapiro sequence and φ(n) = n^2) indicate that there might be a more general principle behind.
Finally, we also consider the subsequences along primes of automatic sequences. It seems that showing normality for such sequences is out of reach. However, it is possible to describe the frequences of letters for these subsequences.

15h10-16h00, Christian Krattenthaler (Université de Wien), Analogues discrets des intégrales de Macdonald-Mehta

Je regarderai certaines discrétisations des célèbres intégrales de Macdonald-Mehta. Il y a dix familles de telles intégrales qui se calculent en formes closes. J'esquisserai les idées des preuves de ces identités. Ils viennent respectivement des identités pour des caractères de groupes classiques, de la combinatoire de chemins deux à deux disjoints, et d'une formule de transformation pour des séries hypergéométriques elliptiques. Il s'agit d'un travail en commun avec Richard Brent and Ole Warnaar.

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