{"paper":{"title":"On balancing consecutive slices of cake","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"David Bevan","submitted_at":"2026-07-01T11:00:53Z","abstract_excerpt":"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 $\\inf\\mu_r(\\boldsymbol{a})$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.00775","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.00775/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}