  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.
