Excluding affine configurations over a finite field

Let $a_{i1}x_1+\cdots+a_{ik}x_k=0$, $i\in[m]$ be a balanced homogeneous system of linear equations with coefficients $a_{ij}$ from a finite field $\mathbb{F}_q$. We say that a solution $x=(x_1,\ldots, x_k)$ with $x_1,\ldots, x_k\in \mathbb{F}_q^n$ is `generic' if every homogeneous balanced linear equation satisfied by $x$ is a linear combination of the given equations. We show that if the given system is `tame', subsets $S\subseteq \mathbb{F}_q^n$ without generic solutions must have exponentially small density. Here, the system is called tame if for every implied system the number of equations is less than half the number of used variables. Using a subspace sampling argument this also gives a `supersaturation result': there is a constant $c$ such that for $\epsilon>0$ sufficiently small, every subset $S\subseteq \mathbb{F}_q^n$ of size at least $q^{(1-\epsilon) n}$ contains $\Omega(q^{(k-m-\epsilon c)n})$ solutions as $n\to\infty$. For $q<4$ the tameness condition can be left out. Our main tool is a modification of the slice rank method to leverage the existence of many solutions in order to obtain high rank solutions.