The goal of this note is to outline Solymosi’s sum–product argument and the first step in the recent Konyagin–Shkredov improvement.

Let be finite. We define the *sumset* and *product set* via

Recall the famous Erdős–Szemerédi sum–product conjecture

**Conjecture (ESconj):** Fix . Then there exists a such that for any finite , one has

♠

We begin with Solymosi’s argument, but first we need to recall a couple definitions. We define two representation functions of :

The *additive energy* and *multiplicative energy* of and are defined via

We set and . Cauchy–Schwarz implies the following relations

In this post, we adopt the convention if for some and if and .

**Theorem (Sol):** Let be finite. Then

♠

Observe that Theorem (Sol) easily implies any is admissible in Conjecture (ESconj).

*Proof:* Wlog, we may assume . Consider . For

we let be the ray

So . Let be a parameter to be chosen later and

Thus is the set of rays emanating from the origin that contain points of . Let

Here are the two key observations.

- The sets are disjoint for .
- Any element in the open cone generated by and ,
can be written uniquely as where and .

Thus

To choose , note that

and so by a dyadic decomposition, we may choose a such that

This yields the desired result.

**Remark:** One can modify the above argument to show

as was observed by Antal Balog. This was recently improved to

for some fixed . One of their main tools was the Konyagin–Shkredov clustering below. ♠

Konyagin and Shkredov recently improved upon Solymosi’s estimate for Conjecture (ESconj). Note that in the proof of Theorem (Sol) we were only able to count sums from *adjacent* lines, and , above. If we could somehow count sums from all pairs, we would obtain the estimate

which is of course too strong to hope for in general. When can we not do this? Well, this argument actually works, unless there is a “collision” of the form

where each tuple lies in . It is easy to check, after relabeling, that we may assume . For ease of exposition, we set

The two equations in (1) together imply

So we are motivated to define

After accounting for collisions, we find that

We cannot use this argument as is, since we have no way to bound the number of collisions for that many . The idea is to partition into disjoint sets of consecutive lines. For , we may apply the above argument to get that the number of elements of in the open cone generated by and is

Summing over disjoint sets of consecutive lines we ultimately find

So we take

where is any quantity satisfying

Thus we have improved Solymosi’s argument from to , as long as

Thus we hope to be able to take to be a power of . The bad cases are when is smaller than some fixed power of and . It turns out to be relatively easy to handle the case , so the key case to analyze is when is smaller than some fixed power of and

In light of the trivial bound

this should impose additive structure on , and . Konyagin and Shkredov were able to successfully analyze this case as well in order to, at the time, obtain further progress towards Conjecture (ESconj).

## 1 thought on “Konyagin-Shkredov Clustering”