矩阵模式的常见差异
Popular differences for matrix patterns
摘要：
下面的组合猜想自然产生于最近的遍历理论工作的阿克尔斯堡，伯格森，最好的。设$M_1$，$M_2$是$k\times k$整数矩阵，$G$是$N$阶的有限阿贝尔群，$A\subseteq G^k$有$|A|\geαN^k$ .如果$M_1$，$M_2$，$M_1-M_2$，和$M_1+M_2$是自同构的$G^k$，是真的，存在一个流行的差异$d \in G^k\setminus\{0\}$，使\[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (α^4-o(1))N^k.\]我们表明，这一猜想是假的一般情况下，但持有$G = \mathbb{F}_p^n$与$p$一个奇素数给定的额外的频谱条件，没有对本征值的$M_1M_2^{-1}$（超过$\overline{\mathbb{F}}_p$）是负的对方。特别是，“旋转的正方形”模式不满足这个特征值条件，我们给出了一个建设的一组正密度在$(\mathbb{F}_5^n)^2$，该模式没有非零流行的差异。这是令人惊讶的对比，三点模式，我们处理所有紧凑的阿贝尔群，不需要一个额外的谱条件。
英文摘要：
The following combinatorial conjecture arises naturally from recent ergodic-theoretic work of Ackelsberg, Bergelson, and Best. Let $M_1$, $M_2$ be $k\times k$ integer matrices, $G$ be a finite abelian group of order $N$, and $A\subseteq G^k$ with $|A|\geαN^k$. If $M_1$, $M_2$, $M_1-M_2$, and $M_1+M_2$ are automorphisms of $G^k$, is it true that there exists a popular difference $d \in G^k\setminus\{0\}$ such that \[\#\{x \in G^k: x, x+M_1d, x+M_2d, x+(M_1+M_2)d \in A\} \ge (α^4-o(1))N^k.\] We show that this conjecture is false in general, but holds for $G = \mathbb{F}_p^n$ with $p$ an odd prime given the additional spectral condition that no pair of eigenvalues of $M_1M_2^{-1}$ (over $\overline{\mathbb{F}}_p$) 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 $(\mathbb{F}_5^n)^2$ 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 an additional spectral condition.
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