Lower bound on certain exponential sums and expressions related to them

Question

Let $$G(\alpha, x) = \sum_{n\le x}e(\alpha n^2)$$ Clearly, $r_k(n)$, the number of representations of a number as the sum of $k$ squares is given by the following expression: $$r_k(n) = \int_0^1 G(\alpha, n)^k e(-\alpha n) d\alpha$$ However, the only bounds that I have seen on $G(\alpha, n)$ and to some extent $r_k(n)$ have been upper bounds. Does anyone know of any nontrivial lower bounds on $r_k(n)$, or asymptotic representations for $G(\alpha, n)$ if $|\alpha - \frac{a}{q}|\le \frac{\ln^B n }{n}$ and $q\le \ln^B n$.

Answer 1

What you are asking corresponds to estimating the exponential sum $$\sum_{n\leq N}e(\alpha n^{2})$$ on the Major arcs $$\mathfrak{M}_{q,a}=\left\{ \alpha\in[0,1]\ :\ \left|\alpha-\frac{a}{q}\right|\leq\frac{1}{N^{1-\eta}}\right\}$$ where $\eta\leq\frac{1}{10}.$ Note that this set is larger than what you are asking for, since I allowed $$\frac{N^{\eta}}{N}\text{ instead of }\frac{\left(\log N\right)^{B}}{N}.$$ I assume you were looking at the latter because that appears as the Major arcs in the proof of Vinogradov's three prime theorem. When working with Waring's problem, we are not constrained by the lack of uniformity in the prime number theorem for arithmetic progressions.

The Hardy-Littlewood Circle method tells us that for any $k\geq6,$ the number of representations of $n$ as a sum of $k$ squares asymptotically equals $$r_{k}(N)=\mathfrak{S}(N)\Gamma\left(\frac{3}{2}\right)^{k}\Gamma\left(\frac{k}{2}\right)^{-1}N^{k/2-1}+O\left(N^{k/2-1-\delta}\right)$$ for an explicit $\delta>0$, where $\mathfrak{G}(N)$ is a constant called the singular series.

As usual let $e(x)=e^{2\pi ix}.$ Then for any $\alpha\in\mathfrak{M}_{q,a}$ $$\sum_{n\leq N}e\left(\alpha n^{2}\right)=\frac{1}{q}S_{a,q}T(\beta)+O\left(x^{2\eta}\right)$$ where $T(\beta)=\frac{1}{2}\sum_{n\leq N}\frac{e(\beta n)}{\sqrt{n}}$ and $S_{a,q}=\sum_{n=1}^{q}e\left(\frac{a}{q}n^{k}\right).$ Note that the asymptotic comes from the Major arcs. The above quantities $T(\beta)$ and $S_{a,q}$ give rise to the so called singular integral $$\int_{0}^{1}T(\beta)^{k}e\left(-\beta x\right)d\beta=\Gamma\left(\frac{3}{2}\right)^{k}\Gamma\left(\frac{k}{2}\right)^{-1}N^{k/2-1},$$ and singular series

$$\mathfrak{S}(N)=\sum_{q=1}^{\infty}\left(\sum_{\begin{array}{c} a=1\\ (a,q)=1 \end{array}}^{q}\left(\frac{S_{a,q}}{q}\right)^{s}e\left(-\frac{a}{q}N\right)\right).$$

There are similar such expressions for writing $n$ as a sum of $s$ $k^{th}$ powers, and I recommend reading about the Hardy-Littlewood circle method applied to Waring's problem to find out more.