Ben Lund, Princeton University

Abstract

A simple construction using axis parallel lines in a N^{1/(d-1)} x ... x N^{1/(d-1)} grid shows that N lines in F^d can determine c_dN^{d/(d-1} joints, for some constant c_d that depends on the dimension d. A slightly more sophisticated construction starts with k hyperplanes in general position, and takes the lines to the intersection of every d-1 hyperplanes. In every dimension d geq 3, this construction gives at least cN^{d/(d-1)} joints for some constant c that does not depend on the dimension. Starting with the work of Guth and Katz, several groups of authors have shown that N lines in F^d determine at most c'_dN^{d/(d-1)} joints, for some constant c'_d>1 that depends on the dimension d. We show that N lines in F^d determine at most N^{d/(d-1)} joints. Removing the dependence of the constant on the dimension is a qualitative improvement over previous bounds, and this result is quantitatively stronger in all dimensions. As with previous results on the problem, the proof uses the polynomial method. In particular, it relies heavily on ideas introduced by Ruxiang Zhang in his work on Carbery's generalized joints conjecture.