On primitive elements of finite fields avoiding affine hyperplanes

Let $n\ge 2$ be an integer and let ${\mathbb{F}}_q$ be the finite field with $q$ elements, where $q$ is a prime power. Given ${\mathbb{F}}_q$-affine hyperplanes ${\mathcal{A}}_1,\dots ,{\mathcal{A}}_n$ of ${\mathbb{F}}_{q^n}$ in general position, we study the existence and distribution of primitive elements of $\mathbb F_{q^n}$,avoidingeach$\mathcal A_i$. We obtain both asymptotic and concrete results, relating to past works on digits over finite fields.