Popular differences for matrix patterns
Aaron J. Berger, Ashwin Sah, Mehtaab S. Sawhney, Jonathan Tidor · Transactions of the American Mathematical Society · 2022
The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let M 1 M_1 , M 2 M_2 be k × k k\times k integer matrices, G G be a finite abelian group of order N N , and A ⊆ G k A\subseteq G^k with | A | ≥ α N k |A|\ge \alpha N^k . If M 1 M_1 , M 2 M_2 , M 1 − M 2 M_1-M_2 , and M 1 + M 2 M_1+M_2 are automorphisms of G k G^k , is it true that there exists a popular difference d ∈ G k