Character sums over integers with restricted $g$-ary digits

Abstract

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 $g\ge 2$. 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.