  We establish a new upper bound for the number of rationals up to a given
height in a missing-digit set, making progress towards a conjecture of
Broderick, Fishman, and Reich. This enables us to make novel progress towards
another conjecture of those authors about the corresponding intrinsic
diophantine approximation problem. Moreover, we make further progress towards
conjectures of Bugeaud--Durand and Levesley--Salp--Velani on the distribution
of diophantine exponents in missing-digit sets.
  A key tool in our study is Fourier $\ell^1$ dimension introduced by the last
named author in [H. Yu, Rational points near self-similar sets,
arXiv:2101.05910]. An important technical contribution of the paper is a method
to compute this quantity.
