# Sharp bounds for the exponential sum of a polynomial

1. A theorem on the growth of exponential sums evaluated at polynomials

Let ${d\geq 2}$ be a positive integer and fix ${\underline{\alpha}=(\alpha_1,\dots,\alpha_d)\in [0,1]^d}$. We are looking at bounds for $\displaystyle S_d(\underline{\alpha}, N):=\sum_{n\leq N}e(\alpha_1n+\cdots +\alpha_d n^d)$

where ${N}$ is a large number. The main theorem in this regard is the following.

Theorem 1

Let ${\epsilon>0}$. Then $\displaystyle |S_d(\underline{\alpha}, N)|\leq N^{1/2+\epsilon},\ \ \ \textrm{as}\ N\longrightarrow +\infty$

for all ${\alpha\in [0,1]^d}$ outside a set of measure tending to 0 with N.

2. A probabilistic model

We will prove such theorem by showing that for any ${\epsilon>0}$ the above exponential sum will have size bigger than $N^{1/2+\epsilon}$ only on a set of measure tending to $0$ with $N$.

Indeed, suppose that ${\underline{\alpha}}$ is now being chosen at random in ${[0,1]^d}$, where the space is equipped with the uniform probability on ${\mathbb{R}^d}$. Let $\displaystyle X_{d,N}=\frac{|S_d(\underline{\alpha}, N)|^2}{N}.$

By expanding the square and using the formula ${\int_{0}^{1}e(\alpha n)d\alpha=1}$ if ${n=0}$ and ${0}$ otherwise,

we may easily deduce that $\displaystyle \mathbb{E}[X_{d,N}^2]=\frac{\#\mathcal{S}(d,N)}{N^2},$

where ${S(d, N)}$ is the set of positive integers solution to the system of equations given by $\displaystyle \left\{ \begin{array}{lll} x_1+x_2=x_3+x_4;\\ \cdots + \cdots = \cdots + \cdots;\\ x_1^d+x_2^d=x_3^d+x_4^d.\end{array} \right.$

This number is actually usually called Vinogradov’s mean value and denoted by ${J_{2,d}(N)}$. Its analytical representation is indeed $\displaystyle J_{2,d}(N)=\int_{[0,1]^d}|\sum_{n\leq N}e(\alpha_1 n+\cdots \alpha_d n^d)|^{4} d\alpha_1\cdots d\alpha_d.$

By a well known theorem of Bourgain, Demeter and Guth we have for any ${\epsilon>0}$ and ${N\geq 2}$ $\displaystyle \#S(d, N)\ll_{d,\epsilon}N^{2+\epsilon}+N^{4-\frac{d(d+1)}{2}+\epsilon}.$

However, here the situation is much easier and we do not need to appeal to such an important theorem. Indeed, since $d\geq 2$, the system has only trivial solutions. More precisely, once fixed a solution $(x_1, x_2, x_3,x_4)$ all the other solutions are just a rearrangement of this tuple. One can see this by noticing that the solutions to the system $\displaystyle \left\{ \begin{array}{lll} x_1+x_2=n_1;\\ \cdots + \cdots = \cdots;\\ x_1^d+x_2^d=n_d.\end{array} \right.$

with $n_1, \dots, n_d$ positive integers determine via Newton’s identities all the elementary symmetric functions in $x_1\dots x_d$. Thus any other solution of it will just be a permutation of the roots of the polynomial $(X-x_1)(X-x_2).$ We conclude that in such case $\#S(d,N) =N^2(2+o(1)).$

Thus, since ${d\geq 2}$ we get ${\mathbb{E}[X_{d,N}^2]\ll 1.}$ Since ${\mathbb{E}[X_{d,N}]=0}$, which is equivalent to ${V(X_{d,N})=\mathbb{E}[X_{d,N}^2]}$, by Chebyshev’s inequality we get $\displaystyle P(|X_{d,N}|>\eta)\ll\frac{1}{\eta^2},\ \textrm{for\ any}\ \eta>0,$ $\displaystyle P(|S_d(\underline{\alpha},N)|>\sqrt{N\eta})\ll \frac{1}{\eta^2}.$

Taking now ${\eta=N^{2\epsilon}}$ and by the arbitrariness of ${\epsilon}$ we deduce that ${|S_d(\underline{\alpha},N)|\leq N^{1/2+\epsilon}}$ for all ${d\geq 2}$ and all ${\alpha\in[0,1]^d}$, apart from those in a certain set of measure at most ${N^{-4\epsilon}}$, tending to ${0}$ with ${N}$.

3. Some further considerations

Remark 1 Considering higher moments of ${X_{d,N}}$ is it possible to prove for instance that $\displaystyle P(|S_d(\underline{\alpha},N)|>1.01\sqrt{N})>0,$

or equivalently, ${S_d(\underline{\alpha},N)}$ deviates a bit from its mean value.

Moreover, as ${d,N\longrightarrow \infty}$ we also have ${X_{d,N}\xrightarrow{\mathscr{L}}Exp(1)}$, as convergence in law.

Some natural questions now arises:

Can the above procedures be made effective?

Can one do better by considering different scales?

Regarding the last one, we just point out that the random variables ${X_{d,N}}$ behave like a random walk. Therefore, under some further assumptions one could use the iterated law of logarithms to deduce that $\displaystyle \limsup_{N\longrightarrow\infty}\frac{X_{d,N}}{\sqrt{N\log\log N}}=1\ \textrm{almost\ surely}.$

Acknowledgement: The above discussion is a slightly enlarged version of what was presented by Sam Chow during one of the usual Number Theory meetings at Warwick. 