Since this paper of Roche–Newton, Rudnev, and Shkredov, the best bound for the sum–product problem in was given by the following.
Theorem (Sum–Product): Let of size at most . Then
The major breakthrough was a point–plane incidence theorem of Rudnev. Let us recall their proof, which has been simplified since their original.
We consider the set of points and lines
Note that each line contains the points and so there are at least incidences. On the other hand, one may apply a point–line incidence, for instance the cartesian version found in Theorem 4 of this paper of Stevens and Zeeuw to obtain an upper bound for the number of incidences between and . Combining these two bounds gives Theorem 1.
We are able to make the following improvement.
Theorem 1: Let of size at most . Then
We can improve this application of a Szemerédi–Trotter type bound in the specific case of the sum–product problem. The basic idea is to evaluate the above proof and obtain a bound for the –rich lines, instead of incidences. We then obtain the bound for the higher order energy: