Thanks to Sophie Stevens for helpful discussions at the Georgia discrete analysis conference leading to this post. We show that if has small (slightly generalized) third order energy, then has a large difference set. We present an argument due to Shkredov and Schoen. This has the advantage of being drastically simpler (and slightly stronger) than the analog for sumsets, which was discussed in this previous blog post. We say if for some .
Let be finite. We define the sumset and product set via
We have the following difference–product conjecture.
Conjecture 1: Let . Then for any finite one has
We say a finite is convex if
The following conjecture is due to Erdős.
Conjecture (Convex): Suppose is convex. Then
Study of the quantity has played a crucial role in recent progress on both conjectures, which we define now. For finite, we define
See my recent paper on Conjecture 1 for a lengthy introduction to . We mention here that and intuitively the larger is the more additive structure has. For instance, we have
An application of Cauchy–Schwarz, as explained in equation 2 of this paper, reveals
Conjecture (dplus): Let be finite. Then
This would imply Conjecture (Convex) and Conjecture 1 for . The goal of the rest of this post is to prove the current state of the art progress due to Shkredov and Schoen. The proof of Theorem 1 is simple enough that we compute the constant.
Theorem 1: Let be a subset of any additive group. Then
as is explained using a simple geometric argument in equation 9 of this paper of Murphy, Rudnev, Shkredov, and Shteinikov. For , let
Now we use Katz–Koelster inclusion of the form
Simplifying gives the (slightly stronger) desired
Developing a fourth moment analog of the above result would be worthwhile as it would have applications to finite field sum-product problem (edit: Ilya Shkredov and I pursued this idea further in a recent preprint).