On the sum of digits of special sequences in finite fields

Abstract

In $$\mathbb {F}_q$$ , Dartyge and Sarkozy introduced the notion of digits and studied some properties of the sum of digits function. We will provide sharp estimates for the number of elements of special sequences of $$\mathbb {F}_q$$ whose sum of digits is prescribed. Such special sequences of particular interest include the set of n-th powers for each $$n\ge 1$$ and the set of elements of order d in $$\mathbb {F}_q^*$$ for each divisor d of $$q-1$$ . We provide an optimal estimate for the number of squares whose sum of digits is prescribed. Our methods combine A. Weil bounds with character sums, Gaussian sums and exponential sums.