On the size of subsets of $\mathbb{F}_q^n$ avoiding solutions to linear systems with repeated columns

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 present general techniques for systems with repeated columns. This class of linear systems is disjoint from the class covered by Sauermann's result, and captures the systems studied by Mimura and Tokushige into a single proof. 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 (where $p \geq k \geq 3$), then $S$ must have exponentially small density.