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Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type

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Abstract

We generalize the construction of Roy’s Fibonacci type numbers to the case of a Sturmian recurrence and we determine the classical exponents of approximation \(\omega _2(\xi )\), \({\widehat{\omega }}_2(\xi )\), \(\lambda _2(\xi )\), \({\widehat{\lambda }}_2(\xi )\) associated with these real numbers. This also extends similar results established by Bugeaud and Laurent in the case of Sturmian continued fractions. More generally we provide an almost complete description of the combined graph of parametric successive minima functions defined by Schmidt and Summerer in dimension two for such Sturmian type numbers. As a side result we obtain new information on the joint spectra of the above exponents as well as a new family of numbers for which it is possible to construct the sequence of the best rational approximations.

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Notes

  1. Note that there is also a construction which gives for every \(\omega \ge 3\) a number \(\xi \) for which \(\omega _2(\xi )=\omega \), but for such \(\xi \) we always have \({\widehat{\omega }}_2(\xi ) = 2\) (see the proof of Theorem 5.1 and Theorem 5.5 of [5]).

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Acknowledgements

I am very grateful to Stéphane Fischler and Damien Roy for giving me a lot of feedback on this paper. I also thank the anonymous referees for their work.

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Correspondence to Anthony Poëls.

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Poëls, A. Exponents of diophantine approximation in dimension 2 for numbers of Sturmian type. Math. Z. 294, 951–993 (2020). https://doi.org/10.1007/s00209-019-02280-2

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