On balancing consecutive slices of cake

Abstract

Let $\boldsymbol{a}=(a_i)_{i=1}^\infty$ 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 $μ^r_n(\boldsymbol{a})$ 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, \[ μ_r(\boldsymbol{a}) \;=\; \limsup_{n\to\infty}μ^r_n(\boldsymbol{a}) , \] can easily be calculated. Hence, for small $r$, we present upper bounds on $\infμ_r(\boldsymbol{a})$.