Biased multilinear maps of abelian groups

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.