  We study the equidistribution of multiplicatively defined sets, such as the
squarefree integers, quadratic non-residues or primitive roots, in sets which
are described in an additive way, such as sumsets or Hilbert cubes. In
particular, we show that if one fixes any proportion less than $40\%$ of the
digits of all numbers of a given binary bit length, then the remaining set
still has the asymptotically expected number of squarefree integers. Next, we
investigate the distribution of primitive roots modulo a large prime $p$,
establishing a new upper bound on the largest dimension of a Hilbert cube in
the set of primitive roots, improving on a previous result of the authors.
Finally, we study sumsets in finite fields and asymptotically find the expected
number of quadratic residues and non-residues in such sumsets, given their
cardinalities are big enough. This significantly improves on a recent result by
Dartyge, Mauduit and S\'ark\"ozy. Our approach introduces several new ideas,
combining a variety of methods, such as bounds of exponential and character
sums, geometry of numbers and additive combinatorics.
