Counting $\beta $ -expansions and the absolute continuity of Bernoulli convolutions

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• 作者：Tom Kempton (1)
  
• 关键词：Beta expansions ; Bernoulli convolutions ; 11A63 ; 28D05 ; 42A85
• 刊名：Monatshefte f眉r Mathematik
• 出版年：2013
• 出版时间：August 2013
• 年：2013
• 卷：171
• 期：2
• 页码：189-203
• 全文大小：203KB
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• 作者单位：Tom Kempton (1)
  
  1. Mathematics Institute, Utrecht University, Budapestlaan 6, 3584 CD, Utrecht, The Netherlands
  
• ISSN：1436-5081

文摘

We study the typical growth rate of the number of words of length $n$ which can be extended to $\beta $ -expansions of $x$ . In the general case we give a lower bound for the growth rate, while in the case that the Bernoulli convolution associated to parameter $\beta $ is absolutely continuous we are able to give the growth rate precisely. This gives new necessary and sufficient conditions for the absolute continuity of Bernoulli convolutions.