The purpose of the post is to recall a theorem of Bourgain and Konyagin that shows cancellation in exponential sums over multiplicative subgroups of , incorporating the point–plane bound incidence bound due to Rudnev. It is notoriously hard to find cancellation in short exponential sums in , for instance improving the Burgess bound is a fundamental open problem in number theory (see this previous blog post for discussion). Bourgain and Konyagin were able to leverage the sum–product phenomenon to show cancellation in certain sums with as few as terms, improving upon the previous best of due to Konyagin (incidentally the Burgess bound relies on a simpler sum–product type bound).
Let be a multiplicative subgroup. We define the Fourier transform of via
In (1) and what follows we adopt the notation of Chapter 4 in Tao and Vu (see also my previous blog post for some basic discussion on the discrete Fourier transform).
For any , we have the trivial bound
which is obtained at the zero frequency. On the other hand, has multiplicative structure and we expect it cannot correlate with an additive character in light of the sum–product phenomenon. This was verified by Bourgain and Konyagin (see also these notes of Green).
Theorem 1 (Bourgain-Glibichuk-Konyagin): Let of size at least . Then
Theorem 1 should be compared to the famous Gauss sum estimate (see for instance this previous blog post), but applies to much smaller multiplicative subgroups. The proof relies on three ideas. The first is that if is large for one , then it is large for many . Indeed it follows from (1) that
We define the spectrum (see Chapter 4 of Tao and Vu for detailed discussion, as well as these notes of Green) via
By Parseval’s identity of the form
and (2), we find
If , this gives
which is only useful for (for instance this works quite well for Gauss sums). Thus we need new ideas to handle the case . Note this is in alignment with the principle that basic Fourier techniques intrinsically have a “square root barrier.”
The second idea we will use is that has little additive structure in the following form. Recall the additive energy of and is defined via
Proposition 1 (Sum–Product): Let and . Then
Proposition 1 should be compared to the trivial bound .
Proof: We will use Rudnev’s point–plane incidence bound (Theorem 3 in this paper). To do so, we note that counts the number of solutions to
Since is a multiplicative subgroup, is the number of solutions to
This is precisely the number of incidences between the point set and planes of the form . Thus by Rudnev’s point–plane incidence bound of the form (note there is a condition on the number of maximum colinnear planes which is trivially satisfied in our cartesian product set–up)
We now move onto the third idea and general principle that is additively structured. The following Lemma is due to Bourgain and can be found in Lemma 4.37 in Tao and Vu.
Lemma 1 (Additive Structure in Spectrum): Let and . Then for any , one has
Lemma 1 roughly asserts that the spectrum is closed under addition. For example, consider the example where . Here is an interval of length (there are more sophisticated examples, see this paper of Green).
Proof of Lemma 1: We set . By assumption we have
for some of modulus 1. By changing the order of summation and Cauchy Schwarz,
Lemma 1 follows from pigeon holing.
Suppose, for the sake of discussion, that for all , does not have additive structure in the strong form form for all . Then we conclude
and so by Lemma 1 we have a significant growth from to . But then applying this again to we have significant growth to and repeating this procedure will eventually contradict the trivial bound
Proof of Theorem 1: Fix be chosen small enough so that is nonempty. By Lemma 1,
Now we use Proposition 1 to provide an upper bound for . By (2), is a union of cosets of , say . Thus by the triangle inequality in and Proposition 1,
Combining with (5), we find
By (3), we find that and so
Now we let , where and . Thus contains more than 0 or immediately conclude Theorem 1. Thus (6) holds for all since increases in size as increases. We have
and so there is a such that
Combining with (6), we find
Choosing , we find that
which is a contradiction for large as long as .
As we saw in the proof, the sum–product phenomenon asserts that has little additive structure which is in tension with the general property that is additively structured.