1,026 research outputs found
On the Hausdorff Dimension of Bernoulli Convolutions
We give an expression for the Garsia entropy of Bernoulli convolutions in
terms of products of matrices. This gives an explicit rate of convergence of
the Garsia entropy and shows that one can calculate the Hausdorff dimension of
the Bernoulli convolution to arbitrary given accuracy whenever
is algebraic. In particular, if the Garsia entropy is not
equal to then we have a finite time algorithm to determine
whether or not .Comment: 23 pages, 2 table
A lower bound for Garsia's entropy for certain Bernoulli convolutions
Let be a Pisot number and let denote Garsia's
entropy for the Bernoulli convolution associated with . Garsia, in 1963
showed that for any Pisot . For the Pisot numbers which
satisfy (with ) Garsia's entropy has been
evaluated with high precision by Alexander and Zagier and later improved by
Grabner, Kirschenhofer and Tichy, and it proves to be close to 1. No other
numerical values for are known.
In the present paper we show that for all Pisot , and
improve this lower bound for certain ranges of . Our method is
computational in nature.Comment: 16 pages, 4 figure
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