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.

Further reading

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).


One Response to “Waring’s Problem: Lecture 3”

  1. Waring’s Problem: Lecture 4 « Theorem of the week Says:

    […] Theorem of the week Expositions of interesting mathematical results « Waring’s Problem: Lecture 3 […]

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