*In which we show that the minor arcs contribute a negligible amount, and study the behaviour at a point in the major arcs.*

- Proposition 5 (The contribution from the minor arcs):
*Suppose that and for some . Then for sufficiently large (depending on and ), we have .*We proved this using our bound from last time on , together with Parseval’s identity (which here comes down to being the statement that ). This result does not give us a good value of ; later in the course, we shall see Hua’s lemma, and this result will enable us to get a much better value of . - Definition of .
- Lemma 6:
*Let be an interval. Then .*We saw that the key idea here is that the value of depends only on the congruence class of modulo , so we partitioned accordingly. - Lemma 7:
*Let be a differentiable function. Then .* - Lemma 8:
*Let be a point in the major arc . Then .*We partitioned into shorter intervals on which we could treat as approximately constant, and then used Lemma 7 to replace the resulting sum by an integral.

#### Further reading

There’s a colourful picture showing major arcs on the Wikipedia page for the Hardy-Littlewood circle method.

#### Preparation for Lecture 5

Next time, we shall put together the contributions from individual points on the major arcs, by integrating. You might like to try this for yourself before the lecture.

Advertisements

May 7, 2012 at 12:17 pm

[…] Theorem of the week Expositions of interesting mathematical results « Waring’s Problem: Lecture 4 […]

May 9, 2012 at 12:10 pm

[…] 13: For sufficiently large (depending on ), we have We saw that this follows immediately from Proposition 5 and Proposition […]

May 21, 2012 at 12:07 pm

[…] 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 , and in Theorem 24 and […]

May 21, 2012 at 12:07 pm

[…] 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 , and in Theorem 24 and […]