Character sums over integers with restricted g-ary digits

We establish upper bounds for multiplicative character sums and exponential sums over sets of integers that are described by various properties of their digits in a fixed base g greater than or equal to 2. Our main tools are the Weil and Vinogradov bounds for character sums and exponential sums. Our results can be applied to study the distribution of quadratic non-residues and primitive roots among these sets of-integers.