Abstract
We define a special set, called resolution space, which corresponds to real numbers obtained via the following resolution rule : every number greater than a given integer a is identified with ά. This space possesses a natural scaling structure and dynamics. We introduce several notions as locking and transient resonance zones, as well as unstable irrationals numbers. This space is the natural object coming in the 1/f frequency noise problem. Special numbers as Markoff’s irrationals are proved to play a specific role. This first criterion must be understood as a finite resolution in space for physical systems.
We then introduce an additional resolution criterion which allows only a finite construction of the previous space. Anatural notion of fuzzy zone is defined. This second criterion is interpreted as a finite time experiment in physics.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Allouche, J-P, Algebraic and analytic randomness, this volume
Nottale, L, Scale-relativity and quantization of the universe I. Theoretical framework, Astron. Astrophys. 327, 867–889, (1997).
Arnold, V,I, Small denominators I: On the mapping of a circle onto itself, Trans. Amer. Math. Soc. Ser. 2, 46 (1965) 213–284.
Arnold, V,I, Remarks on the perturbation theory for problems of Mathieu type, Russian Math. Surveys 38:4 (1983), 215–233.
Yoccoz, J-C, An introduction to small divisors problems, in “From number theory to physics”, Springer-Verlag, New-York, 659–679, (1992).
Cvitanovik, P, Circle maps: irrational winding, in “From number theory to physics”, Springer-Verlag, New-York, (1992).
Hardy, G, Wright, E, An introduction to the theory of numbers, Oxford University Press, Amen House, London E.C.4, (1965).
Kintchine, A. Ya., Continued fractions, P. Noordho. Ltd., Groningen 1963.
G.A. Jones, D. Singerman, K. Wicks, The modular group and generalized Farey graphs, in Groups St Andrews, 1989, London Math. Soc. Lectures Notes 160, 316–338, (1991).
G. Jones, D. Singerman, Maps, hypermaps and triangle groups, in The Grothendieck theory of Dessins d’enfants, London. Math. Soc. Lectures notes series 200, (1994).
Planat, M, Dos Santos, S, Ratier, N, Cresson, J, Perrine, S, Close to resonance interaction of radiofrequency wawes in a Schottky diode mixer: 1/f noise and number theory, in “Quantum 1/f noise and other low frequency fluctuations in electronic devices”, edited by A. Chung and P. Handel, AIP Press (1999) 177–187.
Planat, M, Dos Santos, S, Cresson, J, Perrine, S, 1/f frequency noise in a communication receiver and the Riemann hypothesis, in “15th International Conference on Noise in Physical Systems and 1/f Fluctuations”, edited by C. Surya, Bentham Press (1999) 409–412.
Planat, M, 1/f frequency noise in a communication receiver and the Riemann hypothesis, this volume.
Queffélec, Transcendance des fractions continues de Thue-Morse, Jounal of Number theory 73, 201–211, (1998).
Serre, J-P, Acourse in Arithmetic, Graduate text in Mathematics 7, Springer-Verlag, New-York-Heidelberg-Berlin, 1973.
Series, C, The geometry of Markoff numbers, The Mathematical Intelligencer Vol.7,no.3,20–29, (1985).
Apostol, T, M, Modular functions and Dirichlet series in number theory, Graduate Text in Mathematics 41, Springer-Verlag, 1976.
Perrine, S, Generalized Markoff theories and the associated conformal geometries, this volume.
Cohen, P, On the modular function and its importance in arithmetic, this volume
Haas, A, Diophantine approximation on hyperbolic Riemann surfaces, Acta Mathematica 156, 33–82, (1986).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Cresson, J., Dénarié, JN. (2000). Geometry and Dynamics of Numbers Under Finite Resolution. In: Planat, M. (eds) Noise, Oscillators and Algebraic Randomness. Lecture Notes in Physics, vol 550. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45463-2_15
Download citation
DOI: https://doi.org/10.1007/3-540-45463-2_15
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67572-3
Online ISBN: 978-3-540-45463-2
eBook Packages: Springer Book Archive