Colouring multijoints

Tony Carbery
University of Edinburgh

Let $L_1, \dots , L_n$ be finite sets of lines in $\mathbb{F}^n$ where $\mathbb{F}$ is any field. A {\em multijoint} is a point of $\mathbb{F}^n$ which is at the intersection of a line from each family in such a way that the directions of the lines span. Call the set of multijoints $J$.
We prove (subject to a technical hypothesis) that it is possible to $n$-colour the multijoints in such a way that every line contains at most $O_n(|J|^{1/n})$ of its own colour. We explain with reference to problems from harmonic analysis why this is the natural result in this context.

This is joint work with Stefan Valdimarsson.

Presentation (PDF File)

Back to Workshop III: The Kakeya Problem, Restriction Problem, and Sum-product Theory