MuDeRa

Multiplicativity, Determinism, and Randomness


     


Project

The aim of the Austrian-French joint project MuDeRa ("Multiplicativity, Determinism, and Randomness", 2015-2017) 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-1751-N26) and the ANR (Agence Nationale de la Recherche, ANR-14-CE34-0009).


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 11 researchers and is based in Graz, Linz, Vienna, Bordeaux, Calais, Marseille, and Nancy.

Postdoctoral researchers:

  • Aistleitner Christoph (TU Graz), 1/10/2015-31/5/2016 (Linz)
  • Aloui Karam (Université de Sfax), 1/2/2016-31/8/2016 (Nancy), 1/2/2017-31/8/2017 (Vienna)
  • Hanna Gautier (Université d'Aix-Marseille), 1/10/2016-31/9/2017
  • Lachand Armand (Université d'Aix-Marseille), 1/9/2015-31/8/2016
  • Spiegelhofer Lukas (TU Vienna), 1/4/2015-31/1/2016 (Nancy)

Doctoral students:

  • Adrian Scheerer (TU Graz), 1/9/2015-30/11/2015 and 1/1/2016-31/8/2017
  • Clemens Müllner (TU Vienna - M. Drmota, Université d'Aix-Marseille - J. Rivat), 09/2014-02/2017, working on a problem posed within MuDeRa

Events


Guests (selection)

  • Christoph Aistleitner (Johannes Kepler Universität Linz), Nancy, 26/02-27/02/2015
  • Domingo Gomez Perez (University of Cantabria), Nancy, 03/06-07/06/2015
  • Jeff Shallit (University of Waterloo, Canada), Nancy, 22/10-23/10/2015
  • Guy Barat (Marseille), Graz
  • Gregory Derfel (Ben Gurion University, Beer Sheva, Israel), Graz
  • Manfred Madritsch (Université de Lorraine, Nancy), Graz
  • Mariusz Lemanczyk (Univ. Torun), TU Vienne, 19/2-22/2/2017
  • Shanta Laishram (Indian Statistical Institute, Delhi, India), Bordeaux, 30/6-11/7/2016
  • Lászlo Mérai (Austrian Academy of Science), Marseille, 17/1-23/1/2016
  • Mohammed Mkaouar (Université de Sfax), Marseille, 28/3-2/4/2016
  • András Sárközy (Loránd Eötvös University, Budapest), Marseille, 7/1-27/1/2016

Publications / Preprints / in preparation (explicitely mentioning MuDeRa)

  • C. Aistleitner and L. Pankowski, Large values of L)functions from the Selberg class, J. Math. Anal. Appl. 446 (2017), no. 1, 345-364
  • C. Aistleitner, A. Hinrichs, and D. Rudolf, On the size of the largest empty box amidst a point set, Discr. Appl. Math., to appear.
  • C. Aistleitner, R. Hofer, and G. Larcher, On evil Kronecker sequences and lacunary trigonometric products. Ann. Inst. Fourier, to appear.
  • C. Aistleitner and G. Larcher, Additive energy and irregularities of distribution, Unif. Distrib. Theory, to appear.
  • C. Aistleitner, G. Larcher, and M. Lewko, Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems. With an appendix by Jean Bourgain; Israel J. Math., to appear.
  • C. Aistleitner, F. Pausinger, A.-M. Svane, and R. Tichy, On functions of bounded variation, Math. Proc. Cambridge, to appear, http://arxiv.org/abs/1510.04522.
  • K. Aloui, C. Mauduit and M. Mkaouar, Répartition simultanée de S(n) et S(n+1) dans les progressions arithmétiques simultanées, Ramanujan Journal 42 (1) (2017), 173--197.
  • K. Aloui, C. Mauduit and M. Mkaouar, Somme des chiffres et repartition dans les classes de congruence pour les palindromes ellipséphiques, Acta Math. Hungarica 151 (2) (2017), 409--455.
  • V. Becher, A.-M. Scheerer and T. Slaman, On absolutely normal numbers and their discrepancy estimate, http://arxiv.org/abs/1702.04072.
  • M. Bennett and A.-M. Scheerer, Squares with three nonzero digits, Festschrift in honour of Robert F. Tichy's 6àth birthday, to appear.
  • S. Bettin and B. Martin, On the non-vanishing of certain Dirichlet series, submitted, 16 pages.
  • C. Dartyge and D. Goméz-Peréz, Various approaches for the study of the complexity of some families of pseudorandom subsets, Revista Matematica Iberoamericana, EMS, to appear.
  • C. Dartyge and M. Szalay, On a phenomenon of Turàn concerning the summands of partitions, Acta Math. Hungarica 149 (2016), 375--395.
  • C. Dartyge, K. Gyarmati and A. Sárközy, On irregularities of distribution of binary sequences relative to arithmetic progressions I, general results, Uniform Distribution Theory, to appear.
  • G. Derfel, P. J. Grabner and R. F. Tichy, On the asymptotic behaviour of the zeros of solutions of one functional-differential equation with rescaling, submitted.
  • J.-M. Deshouillers, Yet another footnote to The least non zero digit of n! in base 12, Uniform Distribution Theory 11 (2016), no.2, 163-167.
  • J.-M. Deshouillers, M. Drmota, and C. Müllner, Automatic sequences generated by synchronizing automata fulfill the Sarnak conjecture, Studia Mathematica 231 (1), 83-95, 2015.
  • J.-M. Deshouillers and G. Grekos, Integral points on a very flat convex curve, Alladi60 Festschrift, submitted.
  • J.-M. Deshouillers, L. Habsieger, S. Laishram and B. Landreau, Sums of the digits in bases 2 and 3, Tichy60, Festschrift, to appear.
  • M. Drmota, R. Hofer and G. Larcher, On the discrepancy of Halton-Kronecker sequences, to appear in a special issue on the occasion of the 60th birthday of Robert Tichy.
  • M. Drmota, C. Mauduit and J. Rivat, The Thue-Morse sequence along squares is normal, submitted.
  • M. Drmota, M. Kauers and L. Spiegelhofer, On a conjecture of Cusick concerning the sum of digits of n and n + t, SIAM J. Discrete Math. 30, No. 2 (2016), 621--649.
  • S. Ferenczi and C. Mauduit, On Sarnak's conjecture and Veech's question for interval exchanges, Journal d'Analyse Mathématique, to appear.
  • S. Ferenczi, J.Kulaga-Przymus, M. Lemanczyk and C. Mauduit, Substitutions and M ¨obius disjointness, Contemporary Math. AMS, to appear.
  • M. Madritsch and R. Tichy, Multidimensional Heilbronn sets, submitted, http://arxiv.org/abs/1606.03049
  • M. G. Madritsch, A.-M. Scheerer and R. F. Tichy, Computable Absolutely Pisot Normal Numbers, submitted, http://arxiv.org/abs/1610.06388
  • B. Martin, C. Mauduit and J. Rivat, Propriétés locales des chiffres des nombres premiers, Journal de l'Institut de Mathématiques de Jussieu, to appear.
  • B. Martin, C. Mauduit and J. Rivat, Conditions simultanées sur les chiffres des nombres premiers, in preparation.
  • B. Martin and S. Jaffard, Multifractal analysis of the Brjuno function, submitted, 28 pages.
  • C. Mauduit and J. Rivat, Rudin-Shapiro sequences along squares, Transactions of the American Mathematical Society, to appear.
  • C. Mauduit, J. Rivat and A. Sárközy, On the digits of sumsets, Canadian Journal of Mathematics, to appear.
  • C. Mauduit, H. Montgomery, and J. Rivat, Moments of a Thue-Morse generating function, Journal d’Analyse Mathématique, to appear.
  • C. Müllner and L. Spiegelhofer, Normality of the Thue--Morse sequence along Piatetski-Shapiro sequences II, to appear in Israel J. Math., http://arxiv.org/abs/1511.01671.
  • J. Rivat and I. Shparlinski, Multiples of squares in short intervals. Functiones et Approximatio 54, no.1, 2016, 57-63.
  • A. Scheerer, Normality in Pisot Numeration Systems, Ergodic Theory and Dynamical Systems 37 (2) (2017), 664-672.
  • A. Scheerer, Computable Absolutely Normal Numbers and Discrepancies, http://arxiv.org/abs/1511.03582
  • L. Spiegelhofer and M. Wallner, Divisibility of binomial coefficients by powers of primes, http://arxiv.org/abs/1604.07089 .
  • T. Stoll, Thue-Morse along two polynomial subsequences, Lecture Notes in Computer Science, Combinatorics on Words, Springer, no. 9304, Chapter 5.
  • T. Stoll, On digital blocks of polynomial values and extractions in the Rudin-Shapiro sequence, RAIRO 50 (1) (2016), 93-99.

Talks

  • C. Aistleitner: One hundred years of uniform distribution theory. Hermann Weyl's seminal paper of 1916. Lehrstuhlseminar, Lehrstuhl für Geometrie und Visualisierung, TU München. Munich, 01/2016
  • C. Aistleitner: Pair correlations and additive energy, Conference on Combinatorial and Additive Number Theory, University Graz, 01/2015
  • C. Aistleitner: Extreme values of the Riemann zeta function via the resonance method, Conference on Analytic Number Theory and Related Areas, Research Institute for Mathematical Sciences, University Kyoto, Japan, 11/2015
  • C. Aistleitner: Extreme values of the Riemann zeta function, Salzburg Mathematics Colloquium, University Salzburg, 10/2015
  • C. Aistleitner: One hundred years of uniform distribution theory. Hermann Weyl's foundational paper of 1916. International Conference on Harmonic Analysis and Approximations VI. Tsaghkadzor, Armenia, 09/2015
  • C. Aistleitner: Quasi-Monte Carlo sampling for non-uniform measures, 8th International Congress on Industrial and Applied Mathematics, Beijing, China, 09/2015
  • C. Aistleitner: On parametric Thue-Morse Sequences and Lacunary Trigonometric Products, Conference on Diophantine Approximation and Related Topics, Aarhus, Denmark, 07/2015
  • C. Aistleitner: Low-discrepancy point sets for non-uniform measures, 10th IMACS Seminar on Monte Carlo Methods, Linz, 07/2015
  • C. Aistleitner: GCD sums and the order of the Riemann zeta function, Workshop on Dirichlet Series and Analysis on Polydiscs, NTNU, Trondheim, Norway, 06/2015
  • C. Aistleitner: Discrepancy theory for non-uniform measures, Workshop on Information-Based Complexity. Banach Center, Bedlewo, Poland, 04/2015
  • C. Aistleitner: Quasi-Monte Carlo sampling for non-uniform measures, Kolloquium Stochastik und Versicherungsmathematik, TU Graz, 03/2015
  • C. Aistleitner: Analysis of lacunary trigonometric products, a generalised Duffin-Schaeffer conjecture and discrepancy of sequences in the unit interval, Séminaire de Théorie des Nombres de Nancy-Metz, Institut Élie Cartan, Nancy, 02/2015
  • C. Dartyge: Conference in honor of R. Balasubramanian, Chennai (India), 12/2015
  • C. Dartyge: On the greatest prime factor for the twelfth cyclotomic polynomial, Workshop "New approaches in probabilistic and multiplicative number theory", CRM Montréal, 12/2014
  • J.-M. Deshouillers: On Two Recents Breakthroughs Concerning Prime Numbers, Mathematics Seminar of the School of Physical Sciences of JNU, Delhi, India, 02/2015
  • J.-M. Deshouillers: Sum-free subsets of Z/pZ, Seminar in Number Theory, Zürich, Switzerland, 05/2015
  • J.-M. Deshouillers: Automatic questions concerning the representation of n! in base b, 2nd Workshop on Combinatorics, Number Theory and Dynamical Systems, Rio de Janeiro, 08/2015
  • J.-M. Deshouillers: Questions automatiques relatives à la représentation de n! en base b, Colloque à la mémoire de Pierre Liardet, Chambéry, 09/2015
  • J.-M. Deshouillers: Automatic questions concerning the representation of n! in base b, Number Theory Seminar of the Academy of Sciences, Beijing, China, 10/2015
  • J.-M. Deshouillers: Automatic questions concerning the representation of n! in base b, colloque de théorie des nombres, Chennai (IMSc), 12/2015
  • J.-M. Deshouillers: Mini course and HRI Seminar on automatic sequences, Journées de théorie des nombres (65 ans de Balasubramanian), Allahabad (HRI), 02/2016
  • J.-M. Deshouillers: Sarnak’s conjecture and automatic sequences, Colloque Alladi60, Gainesville FL, 03/2016
  • J.-M. Deshouillers: Sarnak’s conjecture and automatic sequences, "Dynamique et graphes sur les corps finis : aspects algebriques, arithmétiques et algorithmiques", Marseille (CIRM), 03/2016
  • J.-M. Deshouillers: Some aspects of automatic sequences, Bristol Math Seminar, Bristol, UK, 05/2016
  • J.-M. Deshouillers: On the density of some automatic sequences, Math. seminar of the Academy of Sciences. Bratislava, Slovakia, 05/2016
  • J.-M. Deshouillers: Probabilistic methods for Waring’s problem : old and new results, Freiman Fest, Tel Aviv, Israel, 07/2016
  • J.-M. Deshouillers: Integral points on a curve. Math, Seminar of IMSc, Chennai, India, 10/2016
  • J.-M. Deshouillers: Integral points on a convex curve, Math. Seminar of HRI, Allahabad, India, 10/2016
  • J.-M. Deshouillers: Integral points on a convex curve, Math. Seminar of ISI. Delhi, India, 10/2016
  • J.-M. Deshouillers: Sur la différence de la somme des chiffres d’un entier dans les bases 2 et 3. Rencontre de Théorie analytique et élémentaire des nombres à l’IHP. Paris, France, 11/2016
  • J.-M. Deshouillers: Automatic sequences satisfy Sarnak’s conjecture I, Ergodic theory and Moebius disjointness, CIRM, Marseille, 12/2016
  • J.-M. Deshouillers: Sarnak’s conjecture and automatic sequences. Math. seminar of Ashoka University. Sonipat, India, 2/2017.
  • M. Drmota: Special Automatic Sequences and the Sarnak Conjecture, Simon Semester Workshops: Ergodic Theory and Dynamical Systems, Bedlewo (Poland), 11/2015
  • M. Drmota: Subsequences of Automatic Sequences, International Conference in Number Theory, Kerala School of Mathematics, India, 1/2017
  • M. Drmota: The discrepancy of generalized Halton sequences, Festkolloquium zum Anlass des 50. Geburtstags von Herrn Univ.Prof.Dr. Peter Grabner, TU Graz, 05/2016
  • M. Drmota: Normal Subsequences of Automatic Sequences, Ergodic Theory and its Connections with Arithmetics and Combinatorics, CIRM Luminy, 12/2016
  • G. Larcher: Analysis of lacunary trigonometric products and discrepancy of sequences in the unit interval, Colloquium on Harmonic Analysis and Discrepancy Theory, Graz, 03/2015
  • G. Larcher: On parametric Thue-Morse sequences and lacunary trigonometric products, Numeration 2015, Nancy, 05/2015
  • C. Mauduit: Automata and number theory, First mini symposium of the Roman Number Theory Association, Rome, Italie, 05/2015
  • C. Mauduit: Presque-périodicité et pseudo-aléa, 18-ème congrès de l'Institut Universitaire de France, Montpellier, 05/2015
  • C. Mauduit: Normality along squares, Journées de Dynamique 2015, Paris, 10/2015
  • C. Mauduit: Variations sur des thèmes de théorie combinatoire des nombres, Colloque Théorie des Nombres et Applications, Monastir, Tunisie, 10/2015
  • C. Mauduit: Digits and pseudo-randomness, Colloque Dynamics and Graphs over Finite Fields : Algebraic, Number Theoretic and Algorithmic Aspects, CIRM-SMF, Luminy, 03/2016
  • J. Rivat: Digital properties of prime numbers, Mini symposium of the Roman Number Theory Association, Rome, 05/2015
  • J. Rivat: Digital properties of prime numbers, Workshop on Combinatorics, Number Theory and Dynamical Systems, Rio de Janeiro, 08/2015
  • J. Rivat: Sopron, 07/2016
  • J. Rivat: Vienna, 11/2016
  • A. Scheerer: Computable Absolutely Normal Numbers, Advanced Topics Seminar, TU Graz, 11/2015
  • A. Scheerer: Poster presentation Computable Absolutely Normal Numbers, Colloque Liardet, Chambéry, 09/2015
  • A. Scheerer: Normality in Pisot Numeration Systems, Journées Arithmétiques, Debrecen, 06/2015
  • A. Scheerer: Normality in Pisot Numeration Systems, Numeration, Nancy, 05/2015
  • A. Scheerer: Normality in Pisot Numeration Systems, Advanced Topics Seminar, TU Graz, 03/2015
  • L. Spiegelhofer: Normality of the Thue-Morse sequence along Piatetski-Shapiro sequences II, Numeration 2015, Nancy, 05/2015
  • L. Spiegelhofer: On a problem by Cusick concerning the sum of digits of n and n + t, Conference "Automatic sequences", Liège, 05/2015
  • L. Spiegelhofer: Divisibility of binomial coefficients by powers of two, Zahlentheoretisches Kolloquium, TU Graz, 01/2016
  • L. Spiegelhofer : Divisibility of binomial coefficients by powers of two, Arbeitsgemeinschaft Diskrete Mathematik, TU Wien, 01/2016
  • L. Spiegelhofer: Divisibility of binomial coefficients by powers of two, Seminaire de Theorie des Nombres de Nancy--Metz, 03/2016
  • T. Stoll: Thue-Morse along two polynomial subsequence, WORDS, Kiel, 09/2015
  • T. Stoll: Quotients de la somme des chiffres des valeurs polynomiales, Colloque "Théorie des Nombres et Applications", Monastir, 10/2015
  • T. Stoll: Sum of digits and subsequences of the integers, Conference on Number Theory, Weinan, China, 12/2016
  • G. Tenenbaum: Analytic number theory in honor of Krishna Alladi 60th birthday, University of Florida at Gainesville, 03/2016
  • R. Tichy: Normality, Computability and Randomness, Aix-Marseille Université, 04/2015
  • R. Tichy: Dynamical Systems in Number Theory, Université de Lorraine, 05/2015
  • R. Tichy: 30 Years of Cooperation with Attila Pethoe, FAN-Workshop, Admont, 06/2015
  • R. Tichy: Dynamical Systems in Number Theory, Journées Arithmétiques 2015, Debrecen, 07/2015
  • R. Tichy: Arithmetical dynamical systems, Conference on Number Theory, Numeration Systems, Ergodic Theory, Chambéry, 09/2015
  • R. Tichy: Probabilistic limit theorems and the Kadec-Pelczynski theorem in L^p, Analysis-Konferenz, Innsbruck, 10/2015
  • R. Tichy: A problem on normality of subsequences, SFB-Winterschool on Quasi Monte Carlo-methods, Traunkirchen, 12/2015