POPULAR DIFFERENCES FOR MATRIX PATTERNS

The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let M₁, M₂ be k × k integer matrices, G be a finite abelian group of order N, and A ⊆ G^k with |A| ≥ αN^k. If M₁, M₂, M₁ − M₂, and M₁ + M₂ are automorphisms of G^k, is it true that there exists a popular difference d ∈ G^k \ {0} such that #{x ∈ G^k : x, x + M₁d, x + M₂d, x + (M₁ + M₂)d ∈ A} ≥ (α⁴ − o(1))N^k? We show that this conjecture is false in general, but holds for G = Fⁿ_p with p an odd prime given the additional spectral condition that no pair of eigenvalues of M₁M₂⁻¹ (over the algebraic closure F_p) are negatives of each other. In particular, the “rotated squares” pattern does not satisfy this eigenvalue condition, and we give a construction of a set of positive density in (Fⁿ₅ )² for which that pattern has no nonzero popular difference. This is in surprising contrast to three-point patterns, which we handle over all compact abelian groups and which do not require additional spectral conditions.