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RANDOM SERIES IN POWERS OF ALGEBRAIC INTEGERS: HAUSDORFF DIMENSION OF THE LIMIT DISTRIBUTION

Published online by Cambridge University Press:  01 June 1998

STEVEN P. LALLEY
STEVEN P. LALLEY
Affiliation:
Department of Statistics, Mathematical Sciences Building, Purdue University, West Lafayette, Indiana 47907, USA. E-mail: [email protected]
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Abstract

We study the distributions Fθ,p of the random sums [sum ]∞1εnθn, where ε1, ε2, … are i.i.d. Bernoulli-p and θ is the inverse of a Pisot number (an algebraic integer β whose conjugates all have moduli less than 1) between 1 and 2. It is known that, when p=.5, Fθ,p is a singular measure with exact Hausdorff dimension less than 1. We show that in all cases the Hausdorff dimension can be expressed as the top Lyapunov exponent of a sequence of random matrices, and provide an algorithm for the construction of these matrices. We show that for certain β of small degree, simulation gives the Hausdorff dimension to several decimal places.

Type
Notes and Papers
Information
Journal of the London Mathematical Society , Volume 57 , Issue 3 , June 1998 , pp. 629 - 654
Copyright
The London Mathematical Society 1998

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