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A lower bound for Garsia’s entropy for certain Bernoulli convolutions

Part of: Algebraic number theory: global fields Real functions

Published online by Cambridge University Press:  01 May 2010

Kevin G. Hare  and
Nikita Sidorov
Kevin G. Hare
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (email: kghare@math.uwaterloo.ca)
Nikita Sidorov
Affiliation:
School of Mathematics, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom (email: sidorov@manchester.ac.uk)

Abstract

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Let β∈(1,2) be a Pisot number and let Hβ denote Garsia’s entropy for the Bernoulli convolution associated with β. Garsia, in 1963, showed that Hβ<1 for any Pisot β. For the Pisot numbers which satisfy xm=xm−1+xm−2+⋯+x+1 (with m≥2), Garsia’s entropy has been evaluated with high precision by Alexander and Zagier for m=2 and later by Grabner, Kirschenhofer and Tichy for m≥3, and it proves to be close to 1. No other numerical values for Hβ are known. In the present paper we show that Hβ>0.81 for all Pisot β, and improve this lower bound for certain ranges of β. Our method is computational in nature.

MSC classification

Secondary: 26A30: Singular functions, Cantor functions, functions with other special properties 11R06: PV-numbers and generalizations; other special algebraic numbers; Mahler measure
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 130 - 143
Copyright
Copyright © London Mathematical Society 2010

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