Uniformly at Random

Posts Tagged ‘beta expansions

Expansions in irrational bases

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Let \beta be an irrational real number between 1 and 2. Let us write the expansion of 1 in base-\beta:

1 = \sum_{n=1}^\infty a_n \beta^{-n},

where a_n \in \{0,1\}.  (We leave it to the reader to speculate as to why one would do such a perverse thing.)  \beta-expansions are not generally unique.  For example, let \varphi = (1 + \sqrt{5})/2 be the golden ratio.  Then 1 = \varphi^{-1} + \varphi^{-2} and 1 = \sum_{n \geq 2} \varphi^{-n}, so .11 and .01111\cdots are both \varphi-expansions of 1. However, there are certain curious irrational numbers \beta for which 1 has a unique \beta-expansion.  Moreover, there is a even a least \beta between 1 and 2 with this property.  This \beta is the unique solution (approx 1.78723…) to

1 = \sum_{n = 1}^{\infty} t_n \beta^{-n},

where t_n is the number of 1’s mod 2 in the binary expansion of n.  (It seems rather bizarre that the parity of 1’s in the binary expansion of n should have anything to do with this problem!)  This \beta is called the Komornik-Loreti constant.  For the proof of this result see the paper of Komornik and Loreti in the American Mathematical Monthly 105 (1998) 636-639.

Written by uncudh

April 20, 2009 at 3:57 pm

Posted in math

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