Waring’s Problem: Lecture 11

In which we improve various bounds.

  • Proposition 28: Let q \geq 2 be an integer and let a be coprime to q.  Then |S(a,q)| \ll q^{-\frac{1}{k}}.  We defined T(a,q) = q^{\frac{1}{k}}S(a,q) and used the bounds from Lemma 26 and Lemma 27, together with the multiplicativity from Lemma 16.
  • We 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 s to s \geq 2k+1, and in Proposition 23 we could relax it to s \geq 2k + 1 for k odd and s \geq 4k for k even.
  • Lemma 29 (Hua’s lemma): We have
    \displaystyle \int_0^1 |\widehat{1_S}(\theta)|^{2^k} \mathrm{d}\theta \ll_{\epsilon} N^{\frac{1}{k}(2^k-k+\epsilon)}
    for any \epsilon > 0.
      We proved 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 equation.
  • We saw that this enables us to improve some earlier bounds.  In particular, in Proposition 5 we could show that the contribution from the minor arcs is negligible if s \geq 2^k+1, and in Theorem 24 and Corollary 25 the assertions hold for s \geq 2^k+1.

Further reading

Here’s Hua’s original paper.  There’s an interesting biography of Hua here.  I particularly enjoyed the phrase “About the only easy thing about Waring’s problem is its statement”!

Preparation for Lecture 12

The next (and final) lecture is going to be a surprise!

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