## Posts Tagged ‘**beta expansions**’

## Expansions in irrational bases

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

where . (We leave it to the reader to speculate as to why one would do such a perverse thing.) -expansions are not generally unique. For example, let be the golden ratio. Then and , so and are both -expansions of 1. However, there are certain curious irrational numbers for which 1 has a unique -expansion. Moreover, there is a even a **least** between 1 and 2 with this property. This is the unique solution (approx 1.78723…) to

where is the number of 1’s mod 2 in the binary expansion of . (It seems rather bizarre that the parity of 1’s in the binary expansion of should have anything to do with this problem!) This 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.