  This text is devoted to simultaneous approximation to $\xi$ and $\xi^2$ by
rational numbers with the same denominator, where $\xi$ is a non-quadratic real
number. We focus on an exponent $\beta_0(\xi)$ that measures the quality of
such approximations (when they are exceptionally good). We prove that $\beta_0$
takes the same set of values as a combinatorial quantity that measures the
abundance of palindrome prefixes in an infinite word $w$. This allows us to
give a precise exposition of Roy's palindrome prefix method. The main tools we
use are Davenport-Schmidt's sequence of minimal points and Roy's bracket
operation.
