Incidence Bounds on Multijoints and Generic Joints

Abstract

Abstract.A point x ∈ F n is a joint formed by a finite collection L of lines in F n if there exist at least n lines in L through x that span F n . It is known that there are n |L| n n−1 joints formed by L. We say that a point x ∈ F n is a multijoint formed by the finite collections L1, . . . , Ln of lines in F n if there exist at least n lines through x, one from each collection, spanning F n . We show that there are n (|L1| · · · |Ln|) 1 n−1 such points for any field F and n = 3, as well as for F = R and any n ≥ 3.Moreover, we say that a point x ∈ F n is a generic joint formed by a finite collection L of lines in F n if each n lines of L through x form a joint there. We show that, for F = R and any n ≥ 3, there areThis result generalises, to all dimensions, a (very small) part of the main point-line incidence theorem in R 3 in [16] by Guth and Katz.Finally, we generalise our results in R n to the case of multijoints and generic joints formed by real algebraic curves.

On the Number of Rich Lines in High Dimensional Real Vector Spaces

“…The main technique in our proof is the Polynomial Ham Sandwich Theorem which we state below. This poweful tool was invented by Guth and Katz in [2] to give a nearly complete solution to the Erdős distinct distance problem and has been applied, for instance, to give a new proof of the Szemerédi-Trotter theorem, the Pach-Sharir theorem [5] (see [3] for more details) and some variants of the joints problem [4].…”

Joints tightened

“…The following generalization of the joints theorem was conjectured by Carberry and proved in F 3 and R n by Iliopoulou [11,12] and in general F n by Zhang [20]. Zhang also proves a further generalization that counts with multiplicities when a joint is contained in many lines, although we do not discuss it here.…”

“…Zhang also proves a further generalization that counts with multiplicities when a joint is contained in many lines, although we do not discuss it here. Theorem 3.1 (Multijoints [11,12,20]). For every d there is some constant…”

Joints of varieties

“…This variation, known as "multijoints", can be viewed as a discrete analogue of the endpoint multilinear Kakeya problem. The following bound on multijoints was conjectured by Carbery, proved in F 3 and R d by Iliopoulou [26] and in general F d by Zhang [41]. Note that the the multijoints theorem is equivalent to the joints theorem if |L i | are all within a constant factor of each other.…”

“…Since Guth and Katz's original work, there has been significant effort in extending the joints theorem [3,4,5,13,22,23,24,25,26,27,28,39,41]. Kaplan, Sharir, and Shustin [27] and Quilodrán [28] independently extended the joints theorem from R 3 to R d , and these techniques and results extend to arbitrary fields as stated below (also see [3,10,31]).…”