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arXiv · 2607.00775

On balancing consecutive slices of cake

David Bevan
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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 $\mu^r_n(\boldsymbol{a})$ be the ratio between the maximum and minimum sizes of $r$ consecutive pieces. Addressing a question of De Bruijn and Erd\H{o}s, we define a family of sequences for which the asymptotic least upper bound of this ratio, \[ \mu_r(\boldsymbol{a}) \;=\; \limsup_{n\to\infty}\mu^r_n(\boldsymbol{a}) , \] can easily be calculated. Hence, for small $r$, we present upper bounds on $\inf\mu_r(\boldsymbol{a})$.

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David Bevan. 2026-07-01. On balancing consecutive slices of cake. https://arxiv.org/abs/2607.00775

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