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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

$\beta $-expansions with deleted digits
for Pisot numbers $\beta $


Author: Steven P. Lalley
Journal: Trans. Amer. Math. Soc. 349 (1997), 4355-4365
MSC (1991): Primary 11K55, 28A78
DOI: https://doi.org/10.1090/S0002-9947-97-02069-2
MathSciNet review: 1451608
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Abstract: An algorithm is given for computing the Hausdorff dimension of the set(s) $\Lambda =\Lambda (\beta ,D) $ of real numbers with representations $x=\sum _{n=1}^\infty d_n \beta ^{-n} $, where each $d_n \in D $, a finite set of ``digits'', and $\beta >0 $ is a Pisot number. The Hausdorff dimension is shown to be $\log \lambda /\log \beta $, where $\lambda $ is the top eigenvalue of a finite 0-1 matrix $A $, and a simple algorithm for generating $A $ from the data $\beta ,D $ is given.


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Additional Information

Steven P. Lalley
Affiliation: Department of Statistics, Mathematical Sciences Bldg., Purdue University, West Lafayette, Indiana 47907
Email: lalley@stat.purdue.edu

DOI: https://doi.org/10.1090/S0002-9947-97-02069-2
Keywords: Hausdorff dimension, entropy, Pisot number
Received by editor(s): June 12, 1995
Additional Notes: Supported by NSF grant DMS-9307855
Article copyright: © Copyright 1997 American Mathematical Society