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.

Further reading

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}}.)

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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 […]

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