Hi, I have a question about the Hardy-Littlewood method.

Writing $R_s(n)$ for the number of ways to write $n$ as a sum of $s$ $k$-th powers and $f(\alpha )$ for the sum $\sum _{m=1}^Ne(\alpha m^k)$, we have

$R_s(n)=\int _\mathfrak Uf(\alpha )^se(-\alpha n)d\alpha ,$

where $\mathfrak U$ is some unit interval. The aim is to work out an asymptotic expression for $R_s(n)$ by studying this integral. We split the domain of integration into the major and minor arcs:

$R_s(n)=\int _\mathfrak Mf(\alpha )^se(-\alpha n)d\alpha +\int _\mathfrak mf(\alpha )^se(-\alpha n)d\alpha $

and aim to work out an asymptotic expressions for the integral over the major arcs whilst making sure the minor arc contributions are "not too big".

Now, this rests on the fact that $f(\alpha )$ gets "big" near a rational number, and moreover gets "more big" the smaller the denominator of the rational number, so that the major arcs, being intervals around rationals with "small" denominators, give the biggest contribution. My problem essentially is, I think, that I don't understand why this should be true; why does $f$ get "big" near rationals, that is.

In Vaughan's book, we approximate $f(\alpha )$ near these rationals through the function $v(\beta )$ (page 14). I think Lemma 2.7 contains what I don't understand. For example, does this lemma say that

$f(\alpha )\sim \frac {S(q,a)}{q}v(\alpha -a/q)$, as $n\rightarrow \infty $?

And how exactly does it influence our definition of the major arcs and our choice of parameter $v$ in the definition of the major arcs? I assume that $v$ is chosen small enough to gain a saving on the estimate $n^{s/k-1}$ in (2.13) on page 16, and that having $q\leq N^v,\alpha \in \mathfrak M(q,a)$ ensures we get an error no larger than $N^{2v}$ in the lemma, but I still feel I'm missing the big picture in the analysis somehow.

Perhaps I don't have a specific problem as such but want to clarify the situation a little bit; perhaps also it is a problem in that I'm simply not quite settled in the fact that $f(\alpha )$ is large at rationals with small denominators.

In any case, I'd appreciate any thoughts/clarifications/things to think about. Thanks very much.