Finding solutions with distinct variables to systems of linear equations over $\mathbb{F}_p$

Abstract

Let us fix a prime $p$ and a homogeneous system of $m$ linear equations $a_{j,1}x_1+\dots+a_{j,k}x_k=0$ for $j=1,\dots,m$ with coefficients $a_{j,i}\in\mathbb{F}_p$. Suppose that $k\geq 3m$, that $a_{j,1}+\dots+a_{j,k}=0$ for $j=1,\dots,m$ and that every $m\times m$ minor of the $m\times k$ matrix $(a_{j,i})_{j,i}$ is non-singular. Then we prove that for any (large) $n$, any subset $A\subseteq\mathbb{F}_p^n$ of size $|A|> C\cdot \Gamma^n$ contains a solution $(x_1,\dots,x_k)\in A^k$ to the given system of equations such that the vectors $x_1,\dots,x_k\in A$ are all distinct. Here, $C$ and $\Gamma$ are constants only depending on $p$, $m$ and $k$ such that $\Gamma<p$. The crucial point here is the condition for the vectors $x_1,\dots,x_k$ in the solution $(x_1,\dots,x_k)\in A^k$ to be distinct. If we relax this condition and only demand that $x_1,\dots,x_k$ are not all equal, then the statement would follow easily from Tao's slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.