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Random Series in Powers of Algebraic Integers: Hausdorff Dimension of the Limit Distribution
Author(s) -
Lalley Steven P.
Publication year - 1998
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/s0024610798005948
Subject(s) - mathematics , hausdorff dimension , minkowski–bouligand dimension , packing dimension , lyapunov exponent , hausdorff measure , effective dimension , combinatorics , algebraic number , dimension (graph theory) , series (stratigraphy) , sequence (biology) , discrete mathematics , fractal , fractal dimension , mathematical analysis , paleontology , biology , physics , genetics , nonlinear system , quantum mechanics
We study the distributions F θ, p of the random sums∑ 1 ∞ε n θ nwhere ε 1 , ε 2 , … 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, F θ, 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.