  In this paper, we bound the number of solutions to a general Vinogradov
system of equations $x_1^j+\dots+x_s^j=y_1^j+\dots+y_s^j$, $(1\leq j\leq k)$,
as well as other related systems, in which the variables are required to
satisfy digital restrictions in a given base. Specifically, our sets of
permitted digits have the property that there are few representations of a
natural number as sums of elements of the digit set -- the set of squares
serving as a key example. We obtain better bounds using this additive structure
than could be deduced purely from the size of the set of variables. In
particular, when the digits are required to be squares, we obtain diagonal
behaviour with $2k(k+1)$ variables.
