Uniformly de Bruijn sequences and symbolic Diophantine approximation on fractals

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Publication details

DateSubmitted - 25 Jun 2016
DateAccepted/In press - 28 Sep 2016
DateE-pub ahead of print - 27 Apr 2018
DatePublished (current) - 1 Jun 2018
Issue number2
Number of pages23
Pages (from-to)271-293
Early online date27/04/18
Original languageEnglish


Intrinsic Diophantine approximation on fractals, such as the Cantor ternary set, was undoubtedly motivated by questions asked by K. Mahler (1984). One of the main goals of this paper is to develop and utilize the theory of infinite de Bruijn sequences in order to answer closely related questions. In particular, we prove that the set of infinite de Bruijn sequences in $k\geq 2$ letters, thought of as a set of real numbers via a decimal expansion, has positive Hausdorff dimension. For a given $k$, these sequences bear a strong connection to Diophantine approximation on certain fractals. In particular, the optimality of an intrinsic Dirichlet function on these fractals with respect to the height function defined by symbolic representations of rationals follows from these results.

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© 2018 The Author(s)

    Research areas

  • math.CO, math.NT


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