Abstract.
This article is devoted to simultaneous approximation to ξ and ξ2 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 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.
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Fischler, S. Palindromic Prefixes and Diophantine Approximation. Mh Math 151, 11–37 (2007). https://doi.org/10.1007/s00605-006-0425-5
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DOI: https://doi.org/10.1007/s00605-006-0425-5