科研之友
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摘要
Let beta is an element of (1, 2) be a Pisot number and let H(beta) denote Garsia's entropy for the Bernoulli convolution associated with beta. Garsia, in 1963, showed that H(beta)<1 for any Pisot beta. For the Pisot numbers which satisfy x(m)=x(m-1)+x(m-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 H(beta) are known. In the present paper we show that H(beta)>0.81 for all Pisot beta, and improve this lower bound for certain ranges of beta. Our method is computational in nature.
- 出版日期2010
