IJPAM: Volume 52, No. 1 (2009)

A GENERALISED DYADIC NUMBER SYSTEM

E. de Amo$^1$, J. Fernández-Sánchez$^2$
$^{1,2}$Department of Algebra and Mathematical Analysis
University of Almería
Almería, 04120, SPAIN
$^1$e-mail: [email protected]


Abstract.It is defined a representation system for numbers in the unit interval, generalising the dyadic one, and two dynamical systems are given which generate it. Metric results are especially derived from the second of them. The approximative coefficient $\theta _{n}\left( x\right) $ is defined and studied with this second dynamical system. Moreover, it is deduced that, among other results, the Jager pair $\left( \theta _{n},\theta _{n-1}\right) $ has the same distribution on a set of $\lambda $-measure 1, it is concentrated on a denumerable set of segments in $\left[ 0,1\right] ^{2},$ and an explicit expression is given for it.

In addition, Gauss-Kuzmin-Levy and Limit Central Theorem type results are given for some random variables in connection with this representation numbers system.

Received: March 8, 2009

AMS Subject Classification: 26A30, 26A06, 26A09

Key Words and Phrases: dynamical system, dyadic representation system, measure preserving function, ergodicity, entropy, Jager pairs, Bernouillicity, identically distributed random variables

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 52
Issue: 1