## 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 $s \geq 2^k+1$.