Multiplicativity, determinism, and Randomness – MuDeRa
Multiplicativity, Determinism, and Randomness
The aim of the Austrian-French joint project (FWF-ANR) is to make progress on various questions that interrelate and link the notion of randomness (or pseudo-randomness) with the notion of determinism in the context of analytic number theory, combinatorics on words, automata theory, quasi-random sequences and nearby disciplines.
Multiplivative and digital properties of sequences
Many numbertheoretic sequences are per se deterministic but resemble much of the overall behaviour of a random sequence. One important example is the Möbius function. In this context we are interested in studying the independence between the Möbius function and various deterministic functions such as functions that are produced by a dynamical system of zero entropy or by a simple algorithm based on the binary digital representation. Informally, the difficulty of this independence problem reflects the difficulty of the transition from the digital representation of an integer to its multiplicative representation as a product of prime factors. This field of questions is the source of many important open problems in mathematics and computer science. In particular, we mention the construction of normal numbers, the analysis of pseudo-random and complexity measures, and the search for optimal discrepancy estimates for digitally based sequences or other quasi-random sequences which are some of the major lines of research in this joint project.
The methodes mainly employed in the projet are based on those from combinatorics and number theory. In particular, we mention the method of exponential sums, friable numbers, the saddle-point method, generating functions, automatic sequences and the arithmetic, probablisitic and dynamic tools for the study of numeration systems.
As for the research that has already be undertaken within the project (publications and preprints), we would like to mention the following two results:
(1) J.-M. Deshouillers, M. Drmota, and C. Muellner, Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Mathematica 231 (1), 83-95, 2015. In this article, the authors show that Sarnak's conjecture holds true for automatic sequences generated by a symchonizing automaton, as well as a prime number theorem for the associated sequences, In a recent preprint, C. Muellner (see the webpage of the project) shows a Moebius randomness principle for all automatic sequences and a prime number theorem for a large class of automatic sequences.
(2) M. Drmota, C. Mauduit and J. Rivat, The Thue-Morse sequence along squares is normal, soumis. The Thue-Morse sequences is an example of an automatic sequence that is generated by a very simple algorithm. The authors show in this paper that the Thue-Morse sequence is normal along the subsequences of squares : the number of occurrences of any n-tuples of digits is well-distributed.
We mention a few research lines that we will undertake :
- J.-M. Deshouillers, B. Landreau, S. Laishram, C. Mauduit, J. Rivat on the compairison of the sum of digits in base 2 and 3
- B. Martin, C. Mauduit an J. Rivat on local properties of the digits of prime numbers
- B. Martin an M. Balazard on sums associated to the distance to the nearest integer function.
We refer to the webpage of the project :
iecl.univ-lorraine.fr/~Thomas.Stoll/MuDeRa/
Project coordination
Thomas Stoll (Institut Elie Cartan de Lorraine)
The author of this summary is the project coordinator, who is responsible for the content of this summary. The ANR declines any responsibility as for its contents.
Partnership
LMPA Laboratoire de Mathématiques Pures et Appliquées
CNRS DR12_I2M Centre National de la Recherche scientifique délégation Provence et Corse_Institut de Mathématiques de Marseille
TUG Technische Universität Graz
JKUL Johannes Kepler Universität Linz
TUG Technische Universität Graz
IECL Institut Elie Cartan de Lorraine
IMB Institut de Mathématiques de Bordeaux
TUW Technische Universität Wien
Help of the ANR 381,680 euros
Beginning and duration of the scientific project:
September 2014
- 36 Months
