Chapter 4 Spectra of Circumscribed Centres

In Chapter 3 we have shown the circumscribed centres of integer circles of a segment are have non-zero density in $\mathbb{Z}^{2}$ with respect to exhaustive filtration $\mathcal{I}$. Hence, for a given radius $r$, a line segment admits an infinite family of circumscribed circles. In this chapter, we investigate this set of centres of circumscribed circles for a set $S$.