Palindromic Prefixes and Diophantine Approximation

Abstract

This text is devoted to simultaneous approximation to ξ and ξ 2 by rational numbers with the same denominator, where ξ is an irrational non-quadratic real number. We focus on an exponent β 0 (ξ) that measures the regularity of the sequence of all exceptionally precise such approximants. We prove that β 0 (ξ) takes the same set of values as a combinatorial quantity that measures the abundance of palindromic prefixes in an infinite word w. This allows us to give a precise exposition of Roy's palindromic prefix method. The main tools we use are Davenport-Schmidt's sequence of minimal points and Roy's bracket operation.In this Section, we recall the properties [7] of words with many palindromic prefixes, and also the bracket operation introduced by Roy [12].

Palindromic continued fractions

“…The Fibonacci continued fraction ξ ( √ 5−1)/2 occurs in the important work of Roy [21,22]. The reader is directed to [10,13,12] for a detailled study of the standard Sturmian continued fractions, which were proved to be transcendental in [7]. Theorem 2.1 provides an alternative and much shorter proof of the latter result.…”

“…Continued fractions beginning with arbitrarily large palindromes appear in several recent papers [21,22,10,12,4]. Motivated by this and the general problematic mentioned above, we ask whether precocious occurrences of some symmetric patterns in the continued fraction expansion of an irrational real number do imply that the latter is either quadratic, or transcendental.…”

A new exponent of simultaneous rational approximation

“…In [8] Fischler introduced a new exponent of approximation β 0 (ξ) for each real number ξ. When λ(ξ, ξ 2 ) < 1, he defined β 0 (ξ) = +∞.…”

Characterizations of Words with Many Periods

“…We call the palindromic prefixes of the word u Fischler words. Later, in [4] Fischler applied his study to simultaneous approximation to a fixed real number and its square by rational numbers with the same denominator. The following result shows that every extremal FW word is a Fischler word, but not conversely.…”