Chapter 6 Conclusion

In this project we have expanded the theory regarding integer circles in the lattice $\mathbb{Z}^{n}$. We found formulae the density the intersection of integer circles with respect to two exhaustive filtrations, particularly $\mathcal{I}$ and $\mathcal{B}(1,1)$. Our computer-based experiments in Example 3.1.20 suggest that these values should agree with each other.

We found numerical examples for the intersection of three unit integer circles in $\mathbb{Z}^{2}$, and found an approximate example which suggests a nice ratio between the intersection of two unit integer circles and the intersection of one unit integer circle with an integer circle of arbitrary radius.

Continuing from the notion of circumscribed circles (see ), we developed theory relating to the duality of the set of circumscribed centres of a set and the set itself. There is an interesting result here which links to the density, as in this duality we notice that a set $S\subset S_{1}(O)$ has the same set of circumscribed centres as $S_{1}(O)$ if it has a greater density than the intersection of any two integer circles.

We introduced the notion of a bisector of an angle and defined its formula. This notion comes from the notion of an inscribed integer circle of an angle. Interestingly, the centres of inscribed circles of two lines forms a ray. The angle between one of the given lines and the ray is the same as the other angle, which makes this ray of inscribed centres a sensible notion of an angle bisector. This idea does not extend as straight-forwardly to $k$-sectors of integer angle, as a $2:1$ ratio-bisector is not (necessarily) the same as a $3$-sector.

The primary direction for further study is to compute this density from the expression found in this report. This might be shown through equating the expressions found, and proving that this is correct. We expect the result to be a rational function of the zeta function $\zeta(x)$ for some $x$. We also would like to prove that the equalities we guessed in Examples 3.1.15 and 3.1.16 are true (or conversely, find that they are false) and further generalise the formulae for the intersection of three unit integer circles.

The discovery of a bisector of integer angles brings more oppurtunity to develop the theory of integer angles. Particularly, we are aiming to find a formula for $LLS(\alpha/2)$ for a given $\alpha$. Additionaly, we consider the development of the notion of a $k$-sector of integer angles for an integer $k$ through the methodology of inscribed circles.