## Waring’s Problem: Lecture 3

In which we prove Weyl’s inequality, and see how it helps with the minor arcs.

• Theorem 3 (Weyl’s inequality).  We saw the statement last time.  We proved it using induction on the degree of the polynomial, and repeated squaring and application of Cauchy-Schwarz to ‘differentiate’ repeatedly.  Finally, we checked that the bound for linear polynomials gave the answer we wanted.
• Lemma 4: Suppose that $Q \gg N^{\gamma}$ and $T \gg N^{\gamma}$ for some $\gamma > 0$.  If $\theta \in \mathfrak{m}$, then $|\widehat{1_S}(\theta)| = O(N^{1/k - \gamma'})$, where $\gamma' > 0$ depends on $\gamma$.  We proved this using Weyl’s inequality.

Our approach to proving Weyl’s inequality follows Davenport’s book (Analytic methods for Diophantine equations and Diophantine inequalities) very closely.  There is a proof there of the bound $d(m) \ll_{\epsilon} m^{\epsilon}$ that we used in the proof of Weyl’s inequality.  You can also find a proof in these lecture notes by Tim Browning.

#### Preparation for Lecture 4

Next time, we shall conclude our analysis of the minor arcs by showing that, for large enough $s$, their contribution to the integral is negligibly small.  We shall find it useful to know the value of $\int_0^1 |\widehat{1_S}(\theta)|^2 \mathrm{d}\theta$, so you might like to compute that.  (Hint: you should be able to find its exact value, rather than just an upper bound.)

We shall then move on to estimating $\widehat{1_S}(\frac{a}{q} + t)$.

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