Bourgain (2015) estimated the number of prime numbers with a positive
proportion of preassigned digits in base 2. We first present a
generalization of this result to any base g at least 2. We then discuss
a more recent result for the set of squares, which may be seen as one
of the most interesting sets after primes. More precisely, for any
base g, we obtain an asymptotic formula for the number of
squares with a proportion c>0 of preassigned digits. Moreover we
provide explicit admissible values for c depending on g. Our
proof mainly follows the strategy developed by Bourgain for primes in
base 2, with new difficulties for squares. It is based on the circle
method and combines techniques from harmonic analysis together with
arithmetic properties of squares and bounds for quadratic Weyl sums.