## Waring’s Problem: Lecture 5

In which we study the contribution from the major arcs.

• Lemma 9: For any real $\theta$, we have
$\displaystyle \left| \int_0^1 e(\theta x^k) \mathrm{d}x \right| \ll \min(1,|\theta|^{-\frac{1}{k}}).$
This was an exercise in integration.
• Lemma 10: For $s \geq 2k+1$, we have
$\displaystyle \int_{-\infty}^{\infty} \left( \int_0^1 e(\tau x^k) \mathrm{d}x \right)^s e(-\tau) \mathrm{d}\tau = \frac{\Gamma(1 + \frac{1}{k})^s}{\Gamma(\frac{s}{k})},$
where $\Gamma$ is the Gamma function.  This was also an exercise in integration.
• Lemma 11: Let $q$ be a natural number and let $a$ be coprime to $q$.  Then $|S(a,q)| \ll_{\epsilon} q^{-\frac{1}{K} + \epsilon}$ where $K = 2^{k-1}$.  This was an easy consequence of Weyl’s inequality (Theorem 3).
• Proposition 12 (The contribution from the major arcs): Suppose that $Q$ and $T$ satisfy the following conditions:
• $Q \to \infty$ and $T \to \infty$ as $N \to \infty$;
• $Q^{-2} \geq 4TN^{-1}$;
• $Q^{\frac{1}{2}} T^{\frac{1}{2}} \leq N^{\frac{1}{2k}}$;
• $N^{-\frac{1}{2k}} Q^{\frac{5}{2}} T^{\frac{3}{2}} = O(N^{-\gamma'})$ for some $\gamma' > 0$; and
• $Q^2 T^{1-\frac{s}{k}} = O(N^{-\gamma'})$ for some $\gamma'>0$.

(For example, take $Q = N^{\gamma}$ and $T = N^{\gamma}$ for a suitable $\gamma > 0$.) Then for $s \geq 2^k + 1$ we have
$\displaystyle \int_{\mathfrak{M}} \widehat{1_S}(\theta)^s e(-N \theta) \mathrm{d}\theta = \frac{\Gamma(1 + \frac{1}{k})^s}{\Gamma(\frac{s}{k})} N^{\frac{s}{k}-1} \sum_{q=1}^{\infty} \sum_{\substack{ a \pmod{q} \\ (a,q)=1}} S(a,q)^s e(-\frac{aN}{q}) + o(N^{\frac{s}{k} - 1}).$
We started proving this, using our estimate for $\widehat{1_S}(\frac{a}{q} + t)$ from last time and then just chasing through the resulting errors. We’ll finish the proof next time.

The usual collection of books and lecture notes.  Please leave a comment if you find another exposition that you like.

#### Preparation for Lecture 6

You could try to finish the proof of Proposition 12 for yourself.  You could also try to get an improved bound on $|S(a,q)|$.  (Later in the course, we shall prove that if $a$ and $q$ are coprime then $|S(a,q)| \ll q^{-\frac{1}{k}}$.)

### 3 Responses to “Waring’s Problem: Lecture 5”

1. Waring’s Problem: Lecture 6 « Theorem of the week Says:

[…] Theorem of the week Expositions of interesting mathematical results « Waring’s Problem: Lecture 5 […]

2. Waring’s Problem: Lecture 11 « Theorem of the week Says:

[…] noted that this led to improved bounds in a number of results.  In particular, in Proposition 12 and Lemma 15 we were able to relax the bound on to , and in Proposition 23 we could relax it to […]

3. Waring’s Problem: Lecture 11 « Theorem of the week Says:

[…] noted that this led to improved bounds in a number of results.  In particular, in Proposition 12 and Lemma 15 we were able to relax the bound on to , and in Proposition 23 we could relax it to […]