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 .
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
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).