November 2023
Intermediate to advanced
192 pages
3h 29m
English
In this chapter, we are now going to formulate and prove a variant of Theorem 2.2 ((part (2)), namely Theorem 8.12 where we are able to now give quantifications of relations between
and
in Theorem 2.2 ((part (2)).
Theorem 8.12 consists of two parts. Part (1) deals with the case when we force our distinct points to lie on an ellipse and in part (2), we assume that our distinct points have not too large diameter and are not too close to each other.
Here is our result.
Theorem 8.12. The following holds [40]:
(1) Let
be small enough depending on
. There exist
small enough depending on
such the following holds. Let
and be two collections of distinct points in with
and with ...