A lower bound for Garsia’s entropy for certain Bernoulli convolutions
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Bibliographic record
Abstract
Abstract Let β ∈(1,2) be a Pisot number and let H β denote Garsia’s entropy for the Bernoulli convolution associated with β . Garsia, in 1963, showed that H β <1 for any Pisot β . 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 β are known. In the present paper we show that H β >0.81 for all Pisot β , and improve this lower bound for certain ranges of β . Our method is computational in nature.
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|---|---|---|
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