  We study additive double character sums over two subsets of a finite field.
We show that if there is a suitable rational self-map of small degree of a set
$D$, then this set contains a large subset $U$ for which the standard bound on
the absolute value of the character sum over $U$ and any subset $C$ (which
satisfies some restrictions on its size $|C|$) can be improved. Examples of
such suitable self-maps are inversion and squaring. Then we apply this new
bound to trace products and sum-product equations and improve results of the
first author and of Gyarmati and S\'ark\"ozy.
