Palindromic Prefixes and Diophantine Approximation

Details Palindromic Prefixes and Diophantine Approximation Palindromic Prefixes and Diophantine Approximation

2005-09-22

arxiv.org/abs/math/0509508v3

Monatshefte Math. 151 (2007), 11-37

Mathematics

Number Theory

31 pages; major improvements in the redaction ; to appear in Monatshefte Math

Scientific paper

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.

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