{"paper":{"title":"Patterns of Negative Shifts and Beta-Shifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Katherine Moore, Sergi Elizalde","submitted_at":"2015-12-14T19:31:53Z","abstract_excerpt":"The $\\beta$-shift is the transformation from the unit interval to itself that maps $x$ to the fractional part of $\\beta x$. Permutations realized by the relative order of the elements in the orbits of these maps have been studied for positive integer values of $\\beta$ and for real values $\\beta>1$. In both cases, a combinatorial description of the smallest positive value of $\\beta$ needed to realize a permutation is provided. In this paper we extend these results to the case of negative $\\beta$, both in the integer and in the real case. Negative $\\beta$-shifts are related to digital expansions"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1512.04479","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}