  Consider a system of $m$ balanced linear equations in $k$ variables with
coefficients in $\mathbb{F}_q$. If $k \geq 2m + 1$, then a routine application
of the slice rank method shows that there are constants $\beta,\gamma \geq 1$
with $\gamma < q$ such that, for every subset $S \subseteq \mathbb{F}_q^n$ of
size at least $\beta \cdot \gamma^n$, the system has a solution
$(x_1,\ldots,x_k) \in S^k$ with $x_1,\ldots,x_k$ not all equal. Building on a
series of papers by Mimura and Tokushige and on a paper by Sauermann, this
paper investigates the problem of finding a solution of higher non-degeneracy;
that is, a solution where $x_1,\ldots,x_k$ are pairwise distinct, or even a
solution where $x_1,\ldots,x_k$ do not satisfy any balanced linear equation
that is not a linear combination of the equations in the system.
  In this paper, we focus on linear systems with repeated columns. For a large
class of systems of this type, we prove that there are constants $\beta,\gamma
\geq 1$ with $\gamma < q$ such that every subset $S \subseteq \mathbb{F}_q^n$
of size at least $\beta \cdot \gamma^n$ contains a solution that is
non-degenerate (in one of the two senses described above). This class is
disjoint from the class covered by Sauermann's result, and captures the systems
studied by Mimura and Tokushige into a single proof. Moreover, a special case
of our results shows that, if $S \subseteq \mathbb{F}_p^n$ is a subset such
that $S - S$ does not contain a non-trivial $k$-term arithmetic progression
(with $p$ prime and $3 \leq k \leq p$), then $S$ must have exponentially small
density.
