# Upper bound for an exponential sum involving characters of a finite field

MathOverflow Asked by nahila on December 14, 2020

Let $$q = p^n$$ be a prime power, $$alphainmathbb{F}_{q}$$
a primitive element of the finite field $$mathbb{F}_q$$ and denote by $$chi$$ a non-trivial additive character of $$mathbb{F}_{q}$$. Set
$$omega = explbrace { ifrac{2pi} {q – 1} } rbrace$$,
I am looking for an upperbound on the following sum
$$begin{equation} leftvert sum _{k=0} ^{q – 2} omega ^{k^2} chi (alpha ^k ) rightvert. end{equation}$$
If we denote by $$psi _c$$ the multiplicative character of
$$mathbb{F}^* _{q} = mathbb{F}_{q}setminuslbrace 0rbrace =lbrace alpha ^0 ,cdots,alpha^{q-2}rbrace$$ corresponding to $$cinmathbb{F}_q ^*$$
then the sum can be written as
$$begin{equation} leftvert sum _{cinmathbb{F}_q^*} psi _c (c) chi (c) rightvert. end{equation}$$
The second sum looks much like a Gaussian sum over finite fields, however in this one the multiplicative character changes as well.

ps: The multiplicative character is given by
$$psi _{alpha ^l} (alpha ^k ) = omega ^{lk} = explbrace ifrac{2pi}{q-1} lk rbrace$$ .
The additive character corresponding to an element $$ainmathbb{F}_{p^n}$$ is given by
$$chi _a (b) = explbrace ifrac{2pi}{p} tr(ab) rbrace$$ for all $$binmathbb{F}_{p^n}$$,
where the trace $$tr : mathbb{F}_{p^n} rightarrow mathbb{F}_p$$ is defined by
$$begin{equation} tr(a) = a+a^p + cdots + a^{p^{n-1}}. end{equation}$$

I am going to assume that by an additive character you mean

an irreducible representation $$chi_alpha : mathbb{F}^n_q longrightarrow mathbb{C}$$, i.e. a group homomorphism from the additive group $$(mathbb{F}^n_q ,+)$$ to the multiplicative group $$(mathbb{C},*)$$

which we can prove must all take the form $$begin{equation}chi_alpha : beta mapsto expleft( {frac{2pi i leftlangle alpha ,beta rightrangle }{p }} right)end{equation}$$ where $$leftlangle alpha ,beta rightrangle = sum_i alpha_i beta_i$$, see chapter 4 of Tao for a proof of some of these statements and see ch.2 of Serre or ch.2 of Fulton & Harris for a general (non-abelian) overview of the representation theory perspective on characters. The point is the following

If we let $$begin{equation} f(x) = begin{cases} q psi_x(x) & text{if } x neq 0 \ 0 & text{if } x = 0 \ end{cases} end{equation}$$ then the sum you are considering is equal to the Fourier transform of $$f$$ i.e. $$begin{equation} hat{f}(alpha) = frac{1}{q} sum_{c in mathbb{F} _q } f(c) chi_alpha(c) = sum_{c in mathbb{F} _q^* } psi_c (c) chi_alpha(c) end{equation}$$ see definition 4.6 in Tao.

We apply the Hausdorff-Young inequality theorem 4.8 in Tao to get that $$begin{equation} left(sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{p'} right)^{frac{1}{p'}} leq left(sum_{alpha in mathbb{F} _q } |f(alpha)|^pright)^{frac{1}{p}} = qleft( sum_{c in mathbb{F} _q^* } |psi_c (c) |^pright)^{frac{1}{p}} end{equation}$$ where the LHS is the $$l^{q}$$-norm, the RHS is the $$l^p$$-norm, and $$p$$ satisfies the following $$p^{-1} +q^{-1} = 1 land 1 leq pleq 2$$. Plugging in $$p = 2$$ we get that

$$begin{equation} sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{2} leq qsum_{c in mathbb{F} _q^* } |psi_c (c) |^2 end{equation}$$ which is equivalent to saying that $$begin{equation} mathbb{Var}[hat f] = frac{1}{q}sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{2} leq sum_{c in mathbb{F} _q^* } |psi_c (c) |^2leq q-1. end{equation}$$

Finally, if you can prove that at least $$n$$ many $$alpha$$ give a value $$| hat f(a)| geq sqrt b$$ then you get that $$begin{equation} nb +sup_{alpha in mathbb{F} _q }left| hat f(a)right|^2 leq sum_{alpha in S}left| hat f(alpha)right|^{2} + sup_{alpha in mathbb{F} _q }left| hat f(a)right|^2 leq sum_{alpha in mathbb{F} _q }left| hat f(alpha)right|^{2} leq q(q-1) end{equation}$$

which gives you that the maximum value is at most

$$begin{equation} sup_{alpha in mathbb{F} _q }left|sum_{c in mathbb{F} _q^* } psi_c (c) chi_alpha(c) right| = sup_{alpha in mathbb{F} _q }left| hat f(a)right| leq sqrt{q(q-1)-nb} end{equation}$$

Essentially we reduced the problem of finding an upper bound to that of finding a lower bound.

Answered by Pedro Juan Soto on December 14, 2020

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