On balancing consecutive slices of cake

• Published in 2026
• Added on 2026-08-17

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 $\infμ_r(\boldsymbol{a})$.

Links

• https://arxiv.org/abs/2607.00775v2
• https://arxiv.org/pdf/2607.00775v2

Other information

BibTeX entry

@article{Onbalancingconsecutiveslicesofcake,
	key = {Onbalancingconsecutiveslicesofcake},
	type = {article},
	title = {On balancing consecutive slices of cake},
	author = {David Bevan},
	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 Erd{\H{o}}s, 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{\}}){\$}.},
	comment = {},
	date_added = {2026-08-17},
	date_published = {2026-08-17},
	urls = {https://arxiv.org/abs/2607.00775v2,https://arxiv.org/pdf/2607.00775v2},
	collections = {food},
	url = {https://arxiv.org/abs/2607.00775v2 https://arxiv.org/pdf/2607.00775v2},
	year = 2026,
	urldate = {2026-08-17},
	archivePrefix = {arXiv},
	eprint = {2607.00775},
	primaryClass = {math.CO}
}