Overall idea of estimating major arcs in Waring's problem

Question

This is copied from math.SE after a kind comment's suggestion as I am sure people here are very well knowledged in this method :)

I am currently reading Vaughan's "The Hardy-Littlewood Method", and in particular the chapter on the major arcs for Waring's Problem (2.4) - a (essentially) copy (all proofs are the same with more details) can be found here. However, I am a bit lost of the "general idea" among the many lemmas and technical calculations. It seems that it's just a lot of unrelated lemmas that happen to work together and happen to give a nice bound at the end.

Therefore, I am wondering if someone can provide a high level overview of the idea behind estimating the major arc contribution? For example, the idea behind bounding the minor arcs would be

1. Crudely bounding $\int |f(\alpha)|^s d\alpha \ll \left(\sup_{\alpha\in\mathfrak{m}} |f(\alpha)|\right)^{s-2^k} \int_0^1 |f(\alpha)|^{2^k} d\alpha$
2. Using Weyl's Inequality to bound $|f(\alpha)|$ over the minor arcs and hence the first term
3. Using Hua's Lemma to reduce the constant in the exponent for the second term (over the trivial bound $\int_{\mathfrak{m}} |f(\alpha)|^{2^k} d\alpha \ll \int_0^1 \left|\sum_{m=1}^{N} e(\alpha m^k)\right|^{2^k} \ll N^{2^k}$).

Thank you and hope this helps others!

P.S. Vaughan's text seems to be quite dense, and I essentially have to think about every mathematical (and non-mathematical) statement he makes for a while. Is that normal?

Answer 1

The following explanation not only accounts for the treatments in the major arcs of Waring's problem, but also the major arcs in a general situation.

Suppose $F(\alpha)$ is some exponential sum that we wish to extract arithmetical information from. Then our task will be to estimate

$$g(n)=\int_0^1F(\alpha)e(-n\alpha)\mathrm d\alpha.\tag1 $$

Hardy and Littlewood found out that when $\alpha$ is near rational (e.g. $\alpha=a/q+\beta$ for some $(a,q)=1$), $F(\alpha)$ is well approximated by a product of two functions, first dependent on $a$ and $q$ while the other only depends on $\beta$:

$$ F\left(\frac aq+\beta\right)\sim S(q,a)u(\beta)\tag2 $$

provided that $|\beta|$ is very small (e.g. $|\beta|\le1/Q$ for some large $Q$).

Thus, we are motivated to craft the main term of $g(n)$ by summing over contribution of integrals over arcs that are near rationals (i.e. the major arcs):

$$ \mathfrak M(q,a)=\left\{0\le\alpha\le1:\left|\alpha-\frac aq\right|\le\frac1Q\right\} $$

There are infinitely many rationals in $[0,1]$, so we are not going to sum over $\int_{\mathfrak M(q,a)}$ for every rational. Instead, we only estimate the ones that have small denominators (e.g. $q\le P$):

\begin{aligned} g(n) &=\sum_{q\le P}\sum_{\substack{1\le a\le q\\(a,q)=1}}\int_{\mathfrak M(q,a)}F(\alpha)e(-n\alpha)\mathrm d\alpha+\int_{\text{minor arc}} \\ &\approx\sum_{\color{blue}{q\le P}}\sum_{\substack{1\le a\le q\\(a,q)=1}}S(q,a)e(-na/q)\int_\color{blue}{-1/Q}^\color{blue}{1/Q}u(\beta)e(-n\beta)\mathrm d\beta \\ &\approx\underbrace{\sum_{\color{red}{q\ge1}}\sum_{\substack{1\le a\le q\\(a,q)=1}}S(q,a)e(-na/q)}_{\mathfrak S(n)}\underbrace{\int_\color{red}{-1/2}^\color{red}{1/2}u(\beta)e(-n\beta)\mathrm d\beta}_{J(n)} \end{aligned}

where $\mathfrak S(n)$ is the singular series and $J(n)$ is the singular integral, which can be evaluated by extracting combinatorial properties from $u(\beta)$. Therefore, we have $g(n)\approx \mathfrak S(n)J(n)$.

This is the motivation behind all the lemmas emerging in Vaughan's book. Some lemmas are dedicated to deduce (2), and others are intended to estimate the errors emerging from replacing $q\le P$ with $q\ge1$ and replacing $\pm 1/Q$ with $\pm1/2$. I hope this answer can address your concern.