Multijoints and Factorisation

Abstract

We solve the dual multijoint problem and prove the existence of so-called "factorisations" for arbitrary fields and multijoints of $k_j$-planes. More generally, we deduce a discrete analogue of a theorem due in essence to Bourgain and Guth. Our result is a universal statement which describes a property of the discrete wedge product without any explicit reference to multijoints and is stated as follows: Suppose that $k_1+\dots +k_d=n$. There is a constant $C=C(n)$ so that for any field $\mathbb{F}$ and for any finitely supported function $S:{\mathbb{F}}^n\to {\mathbb{R}}_{\ge 0}$, there are factorising functions $s_{k_j}:{\mathbb{F}}^n\times \mathrm{Gr}(k_j,{\mathbb{F}}^n)\to {\mathbb{R}}_{\ge 0}$ such that $$(V_1\wedge \cdots \wedge V_d)S(p{)}^d\le C\prod _{j=1}^d{s}_{k_j}(p,V_j),$$ for every $p\in {\mathbb{F}}^n$ and every tuple of planes $V_j\in \mathrm{Gr}(k_j,{\mathbb{F}}^n)$, and $$\sum _{p\in {\pi }_j}s(p,e({\pi }_j))=||S|{|}_d$$ for every $k_j$-plane ${\pi }_j\subset {\mathbb{F}}^n$, where $e({\pi }_j)\in \mathrm{Gr}(k_j,{\mathbb{F}}^n)$ denotes the translate of ${\pi }_j$ that contains the origin and $\wedge$ denotes the discrete wedge product.

18 pages, references updated