A multi-linear geometric estimate

Charlotte Aten; Alex Iosevich

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arXiv · 2112.00810

A multi-linear geometric estimate

Charlotte Aten·Alex Iosevich

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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 $\mathbb{F}_q^*$.

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Charlotte Aten, Alex Iosevich. 2021-12-01. A multi-linear geometric estimate. https://arxiv.org/abs/2112.00810

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