On the Size of Subsets of $\mathbb{F}_q^n$ Avoiding Solutions to Linear Systems with Repeated Columns

Abstract

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.