In this paper, the following bounds were established.

Let $A,B$ be finite subsets of an abelian group, with $|A|,|B|\le N$. We fix some $G \subset A\times B$. We define $C = \{a+b:(a,b) \in G\},D=\{a+2b:(a,b)\in G\}, X = \{a-b:(a,b)\in G\}$.

1. If $|C|\le N$, then $|X| \le N^{2-1/6}$, and it is possible for $|X| \ge N^{\log(6)/\log(3)} = N^{2-0.369\dots}$ to hold.
2. If $|C|\le N,|D|\le N$, then $|X| \le N^{2-1/4}$, and it is possible for $|X| \ge N^{2-1/2}$ to hold.

I understand these problems were originally studied to better understand Besicovitch sets and Kakeya's conjecture. I am aware of the paper of Dvir which basically settles Kakeya's conjecture for finite fields, but to my understanding, this would not say anything about what can be deduced of $|X|$.

I find this problem of bounding $|X|$ given $|C|$ and $|D|$ to be quite interesting in its own right, so has this problem been studied further? If so, what are the best known bounds?