This is copied from [math.SE](https://math.stackexchange.com/questions/4556836/overall-idea-of-estimating-major-arcs-in-warings-problem) 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](https://www.williamchen-mathematics.info/lnhlmfolder/hlm01.pdf). 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?



Overall idea of estimating major arcs in Waring's problem
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.


