On balancing consecutive slices of cake

Let $\mathit{a}=(a_i{)}_{i=1}^{\mathrm{\infty }}$ be an infinite sequence of points on a circle. The first $n$ of these points cuts the circle into $n$ pieces. For any given $r$, let ${\mu }_n^r(\mathit{a})$ be the ratio between the maximum and minimum sizes of $r$ consecutive pieces. Addressing a question of De Bruijn and Erdős, we define a family of sequences for which the asymptotic least upper bound of this ratio, $${\mu }_r(\mathit{a})=\underset{n\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}{\mu }_n^r(\mathit{a}),$$ can easily be calculated. Hence, for small $r$, we present upper bounds on $\inf {\mu }_r(\mathit{a})$.