A multi-linear geometric estimate

TL;DR

This paper generalizes geometric estimates for sums and products in finite fields to multi-linear forms, showing that large enough sets can realize all nonzero values of these forms under certain conditions.

Contribution

It extends previous bilinear results to multi-linear forms, providing new bounds and conditions for the attainability of all nonzero values in finite field settings.

Findings

Nontrivial bounds for multi-linear forms in finite fields.

Conditions for sets to realize all nonzero values of multi-linear forms.

Examples of sets satisfying the quantitative assumptions.

Abstract

We give a generalization of the geometric estimate used by Hart and the second author in their 2008 work on sums and products in finite fields. Their result concerned level sets of non-degenerate bilinear forms over finite fields, while in this work we prove that if $E\subset\mathbb{F}_q^d$ is sufficiently large and $\varpi$ is a non-degenerate multi-linear form then $\varpi$ will attain all possible nonzero values as its arguments vary over $E$, under a certain quantitative assumption on the extent to which $E$ is projective. We show that our bound is nontrivial in the case that $n=3$ and $d=2$ and construct examples of sets to which this applies. In particular, we give conditions under which every member of $\mathbb{F}_q^*$ belongs to $A\cdot A\cdot A+A\cdot A\cdot A\cdot A\cdot A\cdot A$ where $A$ is a union of cosets of a subgroup of $\mathbb{F}_q^*$.