紧阿贝尔群上的偏差分方程，I：解的模
Partial difference equations over compact Abelian groups, I: modules of solutions
摘要：
考虑紧阿贝尔群$Z$和闭子群$U_1$，\ldots，$U_k \leq Z$。令$\mathbb{T} := \mathbb{R}/\mathbb{Z}$。本文讨论了可测函数$Z\to \mathbb{T}$的两类函数方程。首先，给定$f:Z\to \mathbb{T}$和$w \in Z$，得到的差分函数是\[d_wf(z) := f(z-w) - f(z).\]在这个符号中，我们研究差分方程组\[d_{u_1}\cdots d_{u_k}f \equiv 0 \quad \forall u_1 \in U_1,\ u_2 \in U_2,\ \ldots\ u_k \in U_k.\]的解。其次，我们研究可测函数$f_i:Z\to \mathbb{T}$的元组，使得$f_i$在$U_i$和\[f_1 + \cdots + f_k = 0.\]的平移下不变。对于这些方程，解形成$\mathcal{F}(Z)$或$\mathcal{F}(Z)^k$的子群，其中$\mathcal{F}(Z)$是可测量函数$Z\to \mathbb{T}$模Haar-a. e的组。平等的解的子群依概率收敛闭，且在$Z$的旋转下全局不变，因此它是完全可度量化的$Z$ -模。我们将给出一个递归描述的结构$Z$ -模块相对于解决方案模块的低阶方程的同类。这些结果是作为一类特殊的$Z$ -模的抽象理论的应用而得到的.我们的大部分工作将证明这类模在各种自然运算下是封闭的。知道了这一点，上面的描述就很容易得出结论。上述类型的偏差分方程可以被看作是高维的反问题的极值版本，方向类似于Gowers的一致性范数。我们的方法还提供了一些信息，这个反问题的“稳定”版本，它涉及的功能，其Gowers规范是足够接近最大的。
英文摘要：
Consider a compact Abelian group $Z$ and closed subgroups $U_1$, \ldots, $U_k \leq Z$. Let $\mathbb{T} := \mathbb{R}/\mathbb{Z}$. This paper examines two kinds of functional equation for measurable functions $Z\to \mathbb{T}$. First, given $f:Z\to \mathbb{T}$ and $w \in Z$, the resulting differenced function is \[d_wf(z) := f(z-w) - f(z).\] In this notation, we study solutions to the system of difference equations \[d_{u_1}\cdots d_{u_k}f \equiv 0 \quad \forall u_1 \in U_1,\ u_2 \in U_2,\ \ldots\ u_k \in U_k.\] Second, we study tuples of measurable functions $f_i:Z\to \mathbb{T}$ such that $f_i$ is invariant under translation by $U_i$ and also \[f_1 + \cdots + f_k = 0.\] For these equations, the solutions form a subgroup of $\mathcal{F}(Z)$ or $\mathcal{F}(Z)^k$, where $\mathcal{F}(Z)$ is the group of measurable functions $Z\to \mathbb{T}$ modulo Haar-a.e. equality. The subgroup of solutions is closed under convergence in probability and is globally invariant under rotations of $Z$, so it is a complete metrizable $Z$-module. We will give a recursive description of the structure of this $Z$-module relative to the solution-modules of lower-order equations of the same kind. These results are obtained as applications of an abstract theory of a special class of $Z$-modules. Most of our work will go into showing that this class of modules is closed under various natural operations. Knowing that, the above descriptions follow as easy consequences. Partial difference equations of the above kind can be seen as an extremal version of the inverse problem for the higher-dimensional, directional analogs of Gowers' uniformity norms. Our methods also give some information about the `stability' version of this inverse problem, which concerns functions whose Gowers norm is sufficiently close to being maximal.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
