Geometry and Dynamics of Numbers Under Finite Resolution
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.
KeywordsContinue Fraction Rotation Number Modular Group Irrational Number Diophantine Approximation
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- 1.Allouche, J-P, Algebraic and analytic randomness, this volumeGoogle Scholar
- 3.Arnold, V,I, Small denominators I: On the mapping of a circle onto itself, Trans. Amer. Math. Soc. Ser. 2, 46 (1965) 213–284.Google Scholar
- 5.Yoccoz, J-C, An introduction to small divisors problems, in “From number theory to physics”, Springer-Verlag, New-York, 659–679, (1992).Google Scholar
- 6.Cvitanovik, P, Circle maps: irrational winding, in “From number theory to physics”, Springer-Verlag, New-York, (1992).Google Scholar
- 7.Hardy, G, Wright, E, An introduction to the theory of numbers, Oxford University Press, Amen House, London E.C.4, (1965).Google Scholar
- 8.Kintchine, A. Ya., Continued fractions, P. Noordho. Ltd., Groningen 1963.Google Scholar
- 9.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).Google Scholar
- 10.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).Google Scholar
- 11.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.Google Scholar
- 12.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.Google Scholar
- 13.Planat, M, 1/f frequency noise in a communication receiver and the Riemann hypothesis, this volume.Google Scholar
- 15.Serre, J-P, Acourse in Arithmetic, Graduate text in Mathematics 7, Springer-Verlag, New-York-Heidelberg-Berlin, 1973.Google Scholar
- 17.Apostol, T, M, Modular functions and Dirichlet series in number theory, Graduate Text in Mathematics 41, Springer-Verlag, 1976.Google Scholar
- 18.Perrine, S, Generalized Markoff theories and the associated conformal geometries, this volume.Google Scholar
- 19.Cohen, P, On the modular function and its importance in arithmetic, this volumeGoogle Scholar