In [this paper](https://arxiv.org/abs/math/9906097) by Katz and Tao, 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?


**Update:** I looked at [this survey](https://arxiv.org/abs/math/0010069) by Katz and Tao which was written shortly after the aforementioned paper. On page 8 of the survey, they talk about this problem of understanding partial sumsets and partial difference sets. They say:



> 
> There is a substantial literature on the relative sizes of sum-sets
> $A\_0 + A\_1$ and difference sets $A\_0 − A\_1$ (see e.g. the excellent survey [17]), but much
> less is known about partial sum-sets and difference sets, when one only considers
> a subset $G$ of pairs $A\_0 \times A\_1$.
> 
> 
> 


It has been 20 years since this survey. Can anyone comment to whether the area of partial sumsets has been studied more since then? If not, I'd be interested to hear some speculation to why this is the case.



Bounding size of partial difference sets given size of partial sumsets
Some improvements to the lower bounds appear in


*Lemm, Marius*, [**New counterexamples for sums-differences**](http://dx.doi.org/10.1090/S0002-9939-2015-12603-2), Proc. Am. Math. Soc. 143, No. 9, 3863-3868 (2015). [ZBL1358.42006](https://zbmath.org/?q=an:1358.42006).


In particular for 1, there are now examples with $|X| \geq N^{1.77898}$ and for 2 there are examples with $|X| \geq N^{1.61226}$. They build upon an earlier entropy-theoretic construction of Ruzsa (folklore, not sure if it was published previously to Lemm's paper) that improved on the lower bounds in my original paper with Nets, in particular producing examples to 1 with $|X| \geq N^{1.726}$.


As far as I know there have been no improvement in the upper bounds (unless one also adds more slices to the hypothesis).


The recent paper


*Green, Ben; Ruzsa, Imre Z.*, [**On the arithmetic Kakeya conjecture of Katz and Tao**](http://dx.doi.org/10.1007/s10998-018-0270-z), Period. Math. Hung. 78, No. 2, 135-151 (2019). [ZBL1438.42040](https://zbmath.org/?q=an:1438.42040).


forms some further equivalent forms of the various problems here, and gives some further applications, but does not numerically improve the exponents for the upper and lower bounds. I would imagine these two papers between them would pretty much summarise the current state of the art.


EDIT: There is also a "no-go" result at


*Katz, Nets Hawk*, [**Elementary proofs and the sums differences problem**](https://eudml.org/doc/41781), Collect. Math. 2006, Spec. Iss., 275-280 (2006). [ZBL1213.42084](https://zbmath.org/?q=an:1213.42084).


that shows that the known upper bound techniques can never improve the bound on $|X|$ below $N^{1.5}$ no matter how many slices one assumes to be bounded.


