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 and for some . If , then , where depends on . 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 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 , their contribution to the integral is negligibly small. We shall find it useful to know the value of , 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 .