ArithRand

Arithmetic Randomness


     


Project

The aim of the Austrian-French joint project ArithRand ("Arithmetic Randomness", 2021-2025) 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. It is financially supported by the FWF (Fonds zur Förderung der wissenschaftlichen Forschung, I 4945-N) and the ANR (Agence Nationale de la Recherche, ANR-20-CE91-0006).


Research topics

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.


Members

The Austrian-French Consortium consists of 16 researchers and is based in Graz, Vienna, Bordeaux, Calais, Marseille, Nancy and St Etienne.

Postdoctoral researchers:

Doctoral students:

  • Vincent Gozé (Calais-Littoral)

Events

  • to come

Guests (selection)

  • to come

Publications/Preprints / in preparation (explicitely mentioning ArithRand)

  • C. Aistleitner, B. Borda: Quantum invariants of hyperbolic knots and extreme values of trigonometric products. Preprint, arXiv:2006.08578
  • C. Aistleitner, D. El-Baz, M. Munsch: A pair correlation problem, and counting lattice points with the zeta function. GAFA 31 (2021), no. 3, 483--512
  • C. Aistleitner, S. Baker, N. Technau, N. Yesha: Gap statistics and higher correlations for geometric progressions modulo one. Preprint, arxiv:2010.10355
  • C. Aistleitner, N. Gantert, Z. Kabluchko, J. Prochno, K. Ramanan: Large Deviation Principles for Lacunary Sums. Preprint, axriv:2012.05281
  • C. Aistleitner, B. Borda: Maximizing Sudler products via Ostrowski expansions and cotangent sums. Preprint, arXiv:2104.01379
  • C. Aistleitner, D. El-Baz, M. Munsch: Difference sets and the metric theory of small gaps. Preprint, arXiv:2108.02227
  • C. Aistleitner, B. Borda: A conjecture of Zagier and the value distribution of quantum modular forms. Preprint, arXiv:2110.07407
  • La Bretèche, R.; Dress, F.; Tenenbaum, G., Remarques sur une somme liée à la fonction de Möbius, Mathematika 66, no. 2 (2020), 416--421.
  • La Bretèche, R.; Munsch, M.; Tenenbaum, G., Small Gál sums and applications, J. London Math. Soc. 103}(2021), 336--352.
  • La Bretèche, R. Tenenbaum, G., Remarks on the Selberg--Delange method, Acta Arith. 200.4 (2021), 349-369.
  • La Bretèche, R. Tenenbaum, G., On strong and almost sure local limit theorems for a probabilistic model of the Dickman distribution, Lith. Math. J. 61, n°3 (2021), 301-311.
  • La Bretèche, R. Tenenbaum, G., On the gap distribution of prime factors, prépublication.
  • Fernando Chamizo, Bruno Martin,  The convergence of certain Diophantine series. Journal of Number Theory, to appear.  DOI: 10.1016/j.jnt.2021.04.022
  • Dartyge, C; Feutrie, D.;  Tenenbaum, G., Entiers ultrafriables en progressions arithmétiques, Quart. J. Math. (Oxford) (2021), à paraître.
  • Debruyne, G.; Tenenbaum, G., The saddle point method for general partitions, Indag. Math. 31 (2020), 728-738.
  • M. Drmota, M. Lemańczyk, C. Müllner, J. Rivat; Some recent developments on the Sarnak conjecture, manuscript.
  • Michael Drmota, Clemens Müllner, Lukas Spiegelhofer; Primes as sums of Fibonacci numbers; arXiv:2109.04068
  • Fouvry, É.; Tenenbaum, G., Multiplicative functions in large arithmetic progressions and applications, Trans. Amer. Math. Soc., à paraître.
  • O. Ramare, Notes on restriction theory in the primes, arXiv:2109.10180
  • Tenenbaum, G., A note on the normal largest  gap between prime factors, J. Théor. Nombres Bordeaux 31, no. 3 (2019), 747-749.
  • Tenenbaum, G.;  Verwee, J., Effective Erd\H os--Wintner theorems, Proc. Steklov Inst. Math. 314 (2021), 275--289.

Talks


    by Michael Drmota
  • online, contributed, (Logarithmic) Densities for Automatic Sequences along Primes and Squares AG Diskrete Mathematik, TU Wien,  March 16, 2021
  • online, invited, (Logarithmic) Densities for Automatic Sequences along Primes and Squares One World Numeration Seminar, March 30, 2021
  • in person, contributed, Primes as Sums of Fibonacci Numbers AG Diskrete Mathematik, TU Wien,  October 12, 2021

    by Bruno Martin
    invited online talk in the Conference "Diophantine Problems, Determinism and Randomness" in Cirm, 24th november 2020. Title " Some interactions between number theory and multifractal analysis".

    by O. Ramare
  • O. Ramaré, 4 octobre 2021, Séminaire de Théorie des Nombres de Paris, Polynomes trigonom'etriques arithm'etiques aux points de petites hauteurs, in person
  • 6-17 December 2021, Density Estimates and Additive Combinatorics, brainstorming session with four participants, with the support of ArithRand https://www.i2m.univ-amu.fr/events/density-estimates-and-additive-combinatorics/

    by Andrei Shubin
    in person, contributed, Primes in subsets and exponential sums, AG Diskrete Mathematik, TU Wien,  Nov 9, 2021

  • by G. Tenenbaum
  • Number Theory Web Seminar, 10 novembre 2020. Diophantine Problems, Determinism and Randomness, CIRM, Luminy, 23-27 novembre 2020.
  • Analytic & combinatorial number theory in honour of R. Balasubramanian, IMSc, Chennai 16 mars 2021.