Biased multilinear maps of abelian groups

TL;DR

This paper extends the theory of partition and analytic rank to general abelian groups, showing that biased multilinear maps can be decomposed into sums of simpler maps factoring through prime power structures.

Contribution

It generalizes structure theorems for biased multilinear maps from elementary abelian groups to all finite abelian groups.

Findings

Biased multilinear maps decompose into sums of maps factoring through prime power groups.

Maps with non-negligible bias are sums of boundedly many structured multilinear maps.

The set of possible biases for such maps is characterized.

Abstract

We adapt the theory of partition rank and analytic rank to the category of abelian groups. If $A_1, \dots, A_k$ are finite abelian groups and $\phi : A_1 \times \cdots \times A_k \to \mathbf{T}$ is a multilinear map, where $\mathbf{T} = \mathbf{R}/\mathbf{Z}$, the bias of $\phi$ is defined to be the average value of $\exp(i 2 \pi \phi)$. If the bias of $\phi$ is bounded away from zero we show that $\phi$ is the sum of boundedly many multilinear maps each of which factors through the standard multiplication map of $\mathbf{Z}/q\mathbf{Z}$ for some bounded prime power $q$. Relatedly, if $F : A_1 \times \cdots \times A_{k-1} \to B$ is a multilinear map such that $\mathbf{P}(F = 0)$ is bounded away from zero, we show that $F$ is the sum of boundedly many multilinear functions of a particular form. These structure theorems generalize work of several authors in the elementary abelian case to the arbitrary abelian case. The set of all possible biases is also investigated.