Waring’s Problem: Lecture 2

In which we start in earnest.

  • Definition of various pieces of notation.
  • Lemma 1: We have
    \displaystyle \# \{(x_1, \dotsc, x_s) \in [N^{1/k}]^s : N = x_1^k + \dotsb + x_s^k \} = \int_0^1 \widehat{1_S}(\theta)^s e(-N \theta) \mathrm{d}\theta.
    We saw that this follows either from a quick bit of Fourier analysis, or from just writing out the right-hand side.
  • Definition of the major arcs and minor arcs.
  • Lemma 2: Suppose that Q^{-2} \geq 4 N^{-1} T.  Then the major arcs are disjoint.  We noted that the major arcs have width 2N^{-1}T and that the centres of two distinct major arcs are at distance at least Q^{-2}.
  • Theorem 3 (Weyl‘s inequality): Let
    \displaystyle f(x) = \alpha x^k + \alpha_{k-1} x^{k-1} + \dotsb + \alpha_1 x + \alpha_0
    be a real polynomial of degree k with leading coefficient \alpha.  Suppose that \alpha has a rational approximation \frac{a}{q} where q is positive, a and q are coprime, and | \alpha - \frac{a}{q}| \leq \frac{1}{q^2}.  Then, for any P and any \epsilon > 0, we have
    \displaystyle | \sum_{x=1}^P e(f(x))| \ll_{\epsilon} P^{1+\epsilon} (P^{-1/K} + q^{-1/K} + (\frac{P^k}{q})^{-1/K}),
    where K = 2^{k-1} and the implied constant depends only on k and \epsilon.
    We noted that if P^{\delta} \leq q \leq P^{k-\delta} for some fixed \delta > 0, then we get an improvement over the trivial bound of P.  We shall prove the result next time.

Further reading

The same collection of books and notes as last time.

Gareth Taylor is typing up his lecture notes from this course, and has put them on his website.

Terry Tao has some notes about equidistribution and Weyl’s inequality on his blog, which give another perspective.

Preparation for Lecture 3

Next time, we’ll see how repeated squaring (to get `differentiation’) and the Cauchy-Schwarz inequality can be used to reduce Weyl’s inequality to a problem for linear polynomials, and then we’ll finish that off.  We’ll also see how to use Weyl’s inequality to get a bound on |\hat{1_S}(\theta)| for \theta lying in the minor arcs.

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3 Responses to “Waring’s Problem: Lecture 2”

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

    […] Theorem of the week Expositions of interesting mathematical results « Waring’s Problem: Lecture 2 […]

  2. Waring’s Problem: Lecture 5 « Theorem of the week Says:

    […] Lemma 11: Let be a natural number and let be coprime to .  Then where .  This was an easy consequence of Weyl’s inequality (Theorem 3). […]

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

    […] this inductively, using an inequality that we obtained in our proof of Weyl’s inequality (Proposition 3) and interpreting the resulting integral as counting integer solutions to an […]

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