Palindromic Prefixes and Diophantine Approximation

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.