Partial difference equations over compact Abelian groups, I: modules of solutions

Tim Austin

Published 2013-05-30, updated 2014-09-27Version 3

Consider a compact Abelian group $Z$ and closed subgroups $U_1$, ..., $U_k$. 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 $w in Z$, let $d_wf(z) := f(z-w) - f(z)$. In this notation, we shall study solutions to the system of difference equations $d_{u_1}...d_{u_k}f = 0$ for all $u_1$ in $U_1$, $u_2$ in $U_2$, ... $u_k$ in $U_k$. We will give a recursive description of the structure of the complete, metrizable Z-module of all such solutions, relative to the solution-modules of lower-degree equations of the same type. Second, we study tuples of measurable functions $f_i:Z\to \mathbb{T}$ such that $f_i$ is $U_i$-invariant and $f_1 + ... + f_k = 0$. We find again that the $Z$-module of such tuples admits a recursive description in terms of the solutions to simpler such problems. These results are obtained from an abstract theory of a special class of $Z$-modules, assembled out of modules of functions on quotients and subgroups of $Z$. Our main results are that this class of modules is closed under various natural operations, from which the above descriptions follow easily. The motivation for this work is the problem of setting up a higher-dimensional `directional' analog of the inverse theory for the Gowers uniformity norms. The partial difference equations studied here can be naturally seen as the extremal version of this problem, concerning functions whose directional Gowers norms take the maximum possible value. Their description provides a modest first step towards a general inverse theory. In addition to solving this strict, algebraic version of the inverse problem, our methods also give some information about the slightly relaxed version of the problem, which asks about functions whose Gowers norm is not strictly maximal, but very close to it.