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Geometry and dynamics of numbers under finite resolution. (English) Zbl 1019.37004

Planat, Michel (ed.), Noise, oscillators and algebraic randomness. From noise communication system to number theory. Lectures of a school, Chapelle des Bois, France, April 5-10, 1999. Berlin: Springer. Lect. Notes Phys. 550, 305-323 (2000).
Summary: 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 \(\infty\). 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. A natural notion of fuzzy zone is defined. This second criterion is interpreted as a finite time experiment in physics.
For the entire collection see [Zbl 0979.00040].

MSC:

37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010)
11J70 Continued fractions and generalizations
37N99 Applications of dynamical systems
11K50 Metric theory of continued fractions
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