A joint formed by a family of lines in \mathbb{R}^3 is a point lying in at least 3 non-coplanar lines in the family. The problem of finding the optimal upper bound on the number of joints formed by L lines was first solved by Guth and Katz.
In this talk, we will present an estimate that involves counting each joint with a multiplicity, depending on the number of triples of linearly independent lines through it. This estimate is a discrete analogue of the maximal Kakeya operator conjecture. We will also present an estimate on multijoints. In particular, a multijoint is a joint formed by lines in three distinct families, and thus the multijoints problem constitutes an discrete analogue of the endpoint multilinear Kakeya problem solved by Guth. We will conclude with a discussion on what the above estimates could imply for distinct distance problems.
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