The aim of this paper is to provide, in any given base \(g\geqslant 2\) , an asymptotic formula for the number of squares with a proportion \(c>0\) of preassigned digits, together with explicit admissible values of c depending on g. Our proof involves the circle method using the strategy first developed by Bourgain for primes with preassigned digits in base 2, which we refined and generalised to any base. However, squares are much sparser than prime numbers, which leads us to overcome new substantial difficulties. Our method combines techniques from harmonic analysis together with arithmetic properties of squares and bounds for quadratic Weyl sums.
