Palindrome Prefixes and Diophantine Approximation
نویسنده
چکیده
This text is devoted to simultaneous approximation to ξ and ξ by rational numbers with the same denominator, where ξ is a non-quadratic real number. We focus on an exponent β0(ξ) that measures the quality of such approximations (when they are exceptionally good). We prove that β0(ξ) 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.
منابع مشابه
Palindromic Prefixes and Diophantine Approximation
This text is devoted to simultaneous approximation to ξ and ξ by rational numbers with the same denominator, where ξ is an irrational non-quadratic real number. We focus on an exponent β0(ξ) that measures the regularity of the sequence of all exceptionally precise such approximants. We prove that β0(ξ) takes the same set of values as a combinatorial quantity that measures the abundance of palin...
متن کاملPalindrome Prefixes and Episturmian Words
Let w be an infinite word on an alphabet A. We denote by (ni)i≥1 the increasing sequence (assumed to be infinite) of all lengths of palindrome prefixes of w. In this text, we give an explicit construction of all words w such that ni+1 ≤ 2ni + 1 for all i, and study these words. Special examples include characteristic Sturmian words, and more generally standard episturmian words. As an applicati...
متن کاملSelf-similar fractals and arithmetic dynamics
The concept of self-similarity on subsets of algebraic varieties is defined by considering algebraic endomorphisms of the variety as `similarity' maps. Self-similar fractals are subsets of algebraic varieties which can be written as a finite and disjoint union of `similar' copies. Fractals provide a framework in which, one can unite some results and conjectures in Diophantine g...
متن کاملEnumeration and structure of trapezoidal words
Trapezoidal words are words having at most n+1 distinct factors of length n for every n ≥ 0. They therefore encompass finite Sturmian words. We give combinatorial characterizations of trapezoidal words and exhibit a formula for their enumeration. We then separate trapezoidal words into two disjoint classes: open and closed. A trapezoidal word is closed if it has a factor that occurs only as a p...
متن کاملCharacterizations of Words with Many Periods
In 2001 Droubay, Justin and Pirillo studied the so-called standard episturmian words. In 2006 and 2007 Fischler defined another type of words in the framework of simultaneous diophantine approximation. Both classes of words were described in terms of palindromic prefixes of infinite words. In 2003 the authors introduced words which they called extremal FW words after they had met so-called FW w...
متن کامل