Let us fix a prime p and a homogeneous system of m linear equations aj,1x1+ . . . +aj,kxk=0 for j=1, . . ., m with coefficients aj,i ∈ 𝔽p. Suppose that k ≥ 3m, that a_{j,1}+ . . . +a_{j,k}=0 for j=1,. . . ,m and that every m × m minor of the m × k matrix (a_{j,i})_{j,i} is non-singular. Then we prove that for any (large) n, any subset A ⊆ (𝔽p)n of size |A| greater than C Γn contains a solution (x1, . . .,xk) ∈ Ak to the given system of equations such that the vectors x1, . . .,xk ∈ A are all distinct. Here, C and Γ are constants only depending on p, m and k such that Γ < p. The crucial point here is the condition for the vectors x1, . . .,xk in the solution (x1, . . .,xk) ∈ Ak to be distinct. If we relax this condition and only demand that x1, . . .,xk are not all equal, then the statement would follow easily from Tao's slicerank 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 slicerank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.
