On lines, joints, and incidences in three dimensions

We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R 3 and m of their joints (points incident to at least three non-coplanar lines) is Θ ( m 1 / 3 n ) for m ⩾ n , and Θ ( m 2 / 3 n 2 / 3 + m + n ) for m ⩽ n . (ii) In particular, the number of such incidences cannot exceed O ( n 3 / 2 ) . (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O ( n ) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O ( n 3 / 2 ) , established by Guth and Katz, on the number of joints in a set of n lines in R 3 . We also present some further extensions of these bounds, and give a trivial proof of Bourgainʹs conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9].