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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 ofthe $m\times k$ matrix $(a_{j,i})_{j,i}$ is non-singular. Then we prove thatfor 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 ofequations 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 $x_1,\dots,x_k$ inthe solution $(x_1,\dots,x_k)\in A^k$ to be distinct. If we relax thiscondition and only demand that $x_1,\dots,x_k$ are not all equal, then thestatement would follow easily from Tao's slice rank polynomial method. However,handling the distinctness condition is much harder, and requires a newapproach. While all previous combinatorial applications of the slice rankpolynomial method have relied on the slice rank of diagonal tensors, we use aslice rank argument for a non-diagonal tensor in combination with combinatorialand probabilistic arguments.

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