Random Series in Powers of Algebraic Integers: Hausdorff Dimension of the Limit Distribution |
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Authors: | Lalley Steven P. |
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Affiliation: | Department of Statistics, Mathematical Sciences Building, Purdue University West Lafayette, Indiana 47907, USA. E-mail: lalley{at}stat.purdue.edu |
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Abstract: | We study the distributions F,p of the random sums where 1, 2, ... are i.i.d. Bernoulli-p and is theinverse of a Pisot number (an algebraic integer ßwhose conjugates all have moduli less than 1) between 1 and2. It is known that, when p=.5, F,p is a singular measure withexact Hausdorff dimension less than 1. We show that in all casesthe Hausdorff dimension can be expressed as the top Lyapunovexponent of a sequence of random matrices, and provide an algorithmfor the construction of these matrices. We show that for certainß of small degree, simulation gives the Hausdorffdimension to several decimal places. |
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