# Uniformly at Random

## Expansions in irrational bases

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.