# Exponential Sums over Small Subgroups

The purpose of the post is to recall a theorem of Bourgain and Konyagin that shows cancellation in exponential sums over multiplicative subgroups of ${\mathbb{F}_p}$, incorporating the point–plane bound incidence bound due to Rudnev. It is notoriously hard to find cancellation in short exponential sums in ${\mathbb{F}_p}$, 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 ${p^{\delta}}$ terms, improving upon the previous best of ${ p^{1/4+\epsilon}}$ due to Konyagin (incidentally the Burgess bound relies on a simpler sum–product type bound).

Let ${H \leq \mathbb{F}_p^{\times}}$ be a multiplicative subgroup. We define the Fourier transform of ${H}$ via

$\displaystyle \widehat{1_H}(\xi) : = \frac{1}{p} \sum_{h \in H} e_p(-\xi h) , \ \ \ e_p(\theta) : = e^{2 \pi i \theta / p}. \ \ \ \ \ (1)$

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 ${\xi \in \mathbb{F}_p}$, we have the trivial bound

$\displaystyle |\widehat{1_H}(\xi)| \leq \mathbb{P}(H) : = |H| / p,$

which is obtained at the zero frequency. On the other hand, ${1_H}$ 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 ${H \leq \mathbb{F}_p^{\times}}$ of size at least ${p^{\delta}}$. Then

$\displaystyle \sup_{\xi \in \mathbb{F}_p^{\times}}|\widehat{1_H}(\xi)| \leq \mathbb{P}(H) p^{-\epsilon}. \ \ \ \spadesuit$

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 ${|\widehat{1_H}(\xi) |}$ is large for one ${\xi \in \mathbb{F}_p^{\times}}$, then it is large for many ${\xi \in \mathbb{F}_p^{\times}}$. Indeed it follows from (1) that

$\displaystyle \widehat{1_H}(h \xi) = \widehat{1_H}(\xi) , \ \ h \in H \ \ \ \ \ (2)$

We define the spectrum (see Chapter 4 of Tao and Vu for detailed discussion, as well as these notes of Green) via

$\displaystyle {\rm Spec}_{\alpha}(H) : = \{\xi \in \mathbb{F}_p : |\widehat{1_H}(\xi) | \geq \alpha \mathbb{P}(H) \}.$

By Parseval’s identity of the form

$\displaystyle \sum_{\xi \in \mathbb{F}_p} |\widehat{1_H}(\xi)|^2 = \mathbb{P}(H),$

and (2), we find

$\displaystyle |H| \leq | {\rm Spec}_{\alpha}(H)| \leq \mathbb{P}(H)^{-1} \alpha^{-2}.\ \ \ \ \ (3)$

If ${|H| = p^{\delta}}$, this gives

$\displaystyle \alpha \leq p^{1/2 - \delta},$

which is only useful for ${\delta > 1/2}$ (for instance this works quite well for Gauss sums). Thus we need new ideas to handle the case ${\delta \leq 1/2}$. 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 ${H}$ has little additive structure in the following form. Recall the additive energy of ${A}$ and ${B}$ is defined via

$\displaystyle E^+(A,B) = \{(a,a' , b , b' ) \in A^2 \times B^2 : a + b = a' + b' \}.$

Note that

$\displaystyle E^+(A,B) = \sum_x r_{A- B}(x)^2 , \ \ \ r_{A-B}(x) : = \#\{(a,b) \in A \times B : x = a -b \}.$

Proposition 1 (Sum–Product): Let ${H \leq \mathbb{F}_p^{\times}}$ and ${|H|^2 |A| \leq p^2}$. Then

$\displaystyle E^+(H ,A) \leq |H| |A|^{3/2} \ \ \ \spadesuit$

Proposition 1 should be compared to the trivial bound ${E^+(H,A) \leq |H| |A|^2}$.

Proof: We will use Rudnev’s point–plane incidence bound (Theorem 3 in this paper). To do so, we note that ${E^+(H,A)}$ counts the number of solutions to

$\displaystyle h + a = h' + a' , \ \ \ a , a' \in A , \ h, h' \in H.$

Since ${H}$ is a multiplicative subgroup, ${|H|^2 E^+(H,A)}$ is the number of solutions to

$\displaystyle hh_1 + a = h' h_1' + a' , \ \ \ a , a' \in A , \ h, h' , h_1 , h_1' \in H.$

This is precisely the number of incidences between the point set ${H \times H \times A}$ and planes of the form ${h x + a = h' y + z}$. 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)

$\displaystyle I(P , \Pi) \ll n^{3/2} , \ \ \ |P| = |\Pi| = n,$

we find

$\displaystyle |H|^2 E^+(H,A) \ll |H|^3 |A|^{3/2} . \ \ \ \spadesuit$

We now move onto the third idea and general principle that ${{\rm Spec}_{\alpha}(H)}$ 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 ${A \subset \mathbb{F}_p}$ and ${0 < \alpha \leq 1}$. Then for any ${S \subset {\rm Spec}_{\alpha}(A)}$, one has

$\displaystyle \#\{ (\xi_1 , \xi_2) \in S \times S : \xi_1 - \xi_2 \in {\rm Spec}_{\alpha^2/2}(A) \} \geq \frac{\alpha^2}{2} |S|^2. \ \ \ \spadesuit$

Lemma 1 roughly asserts that the spectrum is closed under addition. For example, consider the example ${A = \{1 , \ldots , K\}}$ where ${K = o(p)}$. Here ${{\rm Spec}_{\alpha}(A)}$ is an interval of length ${\asymp \alpha^{-1}}$ (there are more sophisticated examples, see this paper of Green).

Proof of Lemma 1: We set ${f = 1_A}$. By assumption we have

$\displaystyle \alpha \mathbb{P}(A) |S| \leq \sum_{\xi \in S} |\widehat{f}(\xi)| = \sum_{\xi \in S} c(\xi) \widehat{f}(\xi) = \frac{1}{p} \sum_{\xi \in S} \sum_{a \in A} c(\xi) e_p(\xi a),$

for some ${c(\xi) \in \mathbb{C}}$ of modulus 1. By changing the order of summation and Cauchy Schwarz,

$\displaystyle \mathbb{P}(A)^2 \alpha^2 |S|^2 \leq \frac{|A|}{p^2} \sum_{a\in A} \sum_{\xi , \xi' \in S} c(\xi) c(\xi') e((\xi - \xi') a) \leq \mathbb{P}(A) \sum_{\xi , \xi' \in S} |\widehat{f}(\xi - \xi')|.$

Lemma 1 follows from pigeon holing. ${\spadesuit}$

Suppose, for the sake of discussion, that for all ${\alpha}$, ${S \subset {\rm Spec}_{\alpha}(A)}$ does not have additive structure in the strong form form ${r_{S-S}(x) \ll 1}$ for all ${x}$. Then we conclude

$\displaystyle \{ (\xi_1 , \xi_2) \in S \times S : \xi_1 - \xi_2 \in T \} \ll |T|,\ \ \ \ \ (4)$

and so by Lemma 1 we have a significant growth from ${{\rm Spec}_{\alpha}(A)}$ to ${{\rm Spec}_{\alpha^2/2}(A)}$. But then applying this again to ${{\rm Spec}_{\alpha^2/2}(A)}$ we have significant growth to ${{\rm Spec}_{\alpha^4/8}(A)}$ and repeating this procedure will eventually contradict the trivial bound

$\displaystyle |{\rm Spec}_{\alpha}(A)| \leq p.$

When ${H}$ is a multiplicative subgroup, we can show a weaker version of (4) using that ${{\rm Spec}_{\alpha}(H)}$ is a union of cosets of ${H}$ via (2) and Proposition 1. We turn to the details.

Proof of Theorem 1: Fix ${0 < \alpha \leq 1}$ be chosen small enough so that ${S_{\alpha} : = {\rm spec}_{\alpha}(H)}$ is nonempty. By Lemma 1,

$\displaystyle \#\{ (\xi_1 , \xi_2) \in S_{\alpha} \times S_{\alpha} : \xi_1 - \xi_2 \in {\rm Spec}_{\alpha^2/2}(H) \} \geq \frac{\alpha^2}{2} |S_{\alpha}|^2,$

that is

$\displaystyle \frac{\alpha^4}{4} |S_{\alpha}|^4\leq \left( \sum_{z \in {\rm Spec}_{\alpha^2/2}((A)(H) } r_{S_{\alpha} - S_{\alpha}}(z) \right)^2 \leq | {\rm Spec}_{\alpha^2/2}(H)| E^+(S_{\alpha} , S_{\alpha}).\ \ \ \ \ (5)$

Now we use Proposition 1 to provide an upper bound for ${E^+(S_{\alpha} , S_{\alpha})}$. By (2), ${S'_{\alpha} : = S_{\alpha} \setminus \{0\}}$ is a union of cosets of ${H}$, say ${S'_{\alpha} = \cup_{x \in C} Hx}$. Thus by the triangle inequality in ${\ell^2}$ and Proposition 1,

$\displaystyle E^+(S'_{\alpha} )^{1/2} \leq \sum_{x \in C} E^+(S'_{\alpha} , Hx)^{1/2} = \sum_{x \in C} E^+(S'_{\alpha}x^{-1} , H)^{1/2} \ll \frac{|S_{\alpha}|}{|H|}|H|^{1/2} |S'_{\alpha}|^{3/4}.$

Combining with (5), we find

$\displaystyle \alpha^4 |H| | {\rm Spec}_{\alpha}(H)|^{1/2} \ll | {\rm Spec}_{\alpha^2/2}(H)| .$

By (3), we find that ${|{\rm Spec}_{\alpha}(H)| > |H| + 1}$ and so

$\displaystyle \alpha^4 p^{\delta / 2} \leq \alpha^4 |H|^{1/2} \ll \frac{| {\rm Spec}_{\alpha^2/2}(H)|}{|{\rm Spec}_{\alpha}(H)|} .\ \ \ \ \ (6)$

Now we let ${1 > \alpha_1 > \alpha_2 > \ldots > \alpha_{J+1} > 0}$, where ${\alpha_{i+1} = \alpha_i^2 / 2}$ and ${\alpha_1 = p^{-\epsilon}}$. Thus ${{\rm Spec}_{\alpha_1}(H)}$ contains more than 0 or immediately conclude Theorem 1. Thus (6) holds for all ${\alpha_i}$ since ${{\rm Spec}_{\alpha_i}(H)}$ increases in size as ${i}$ increases. We have

$\displaystyle \prod_{i = 1}^{J} \frac{| {\rm Spec}_{\alpha_{i+1}}(H)|}{|{\rm Spec}_{\alpha_i}(H)|} = \frac{| {\rm Spec}_{\alpha_{J+1}}(H)|}{|{\rm Spec}_{\alpha_1}(H)|} \leq p ,$

and so there is a ${1 \leq j \leq J}$ such that

$\displaystyle \frac{| {\rm Spec}_{\alpha_{j+1}}(H)|}{|{\rm Spec}_{\alpha_j}(H)|} \leq p^{1/J}.$

Combining with (6), we find

$\displaystyle p^{- 2^J \epsilon + \delta /2} \frac{1}{2^{J}} \ll \alpha_j^4 p^{\delta/2} \ll p^{1/J}.$

Choosing ${2^J \epsilon \asymp 1}$, we find that

$\displaystyle p^{\delta / 2} \ll \epsilon^{-1} p^{1 / \log \epsilon^{-1}},$

which is a contradiction for ${p}$ large as long as ${\delta \ll \log^{-1} \epsilon^{-1}}$. ${\spadesuit}$

As we saw in the proof, the sum–product phenomenon asserts that ${H}$ has little additive structure which is in tension with the general property that ${{\rm Spec}_{\alpha}(A)}$ is additively structured.