Uniformly de Bruijn sequences and symbolic Diophantine approximation on fractals

Lior Fishman, Keith Merrill, David Simmons

Research output: Contribution to journalArticlepeer-review


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.
Original languageEnglish
Pages (from-to)271-293
Number of pages23
Issue number2
Early online date27 Apr 2018
Publication statusPublished - 1 Jun 2018

Bibliographical note

© 2018 The Author(s)


  • math.CO
  • math.NT

Cite this