The modified prime sieve for primitive elements in finite fields

Let $r \geq 2$ be an integer, $q$ a prime power and $\mathbb{F}_{q}$ the finite field with $q$ elements. Consider the problem of showing existence of primitive elements in a subset $\mathcal{A} \subseteq \mathbb{F}_{q^r}$. We prove a sieve criterion for existence of such elements, dependent only on an estimate for the character sum $\sum_{\gamma \in \mathcal{A}}\chi(\gamma)$. The flexibility and direct applicability of our criterion should be of considerable interest for problems in this field. We demonstrate the utility of our result by tackling a problem of Fernandes and Reis (2021) with $\mathcal{A}$ avoiding affine hyperplanes, obtaining significant improvements over previous knowledge.