{"paper":{"title":"Invariant measures, matching and the frequency of 0 for signed binary expansions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Charlene Kalle, Karma Dajani","submitted_at":"2017-03-18T18:57:05Z","abstract_excerpt":"We introduce a parametrised family of maps $\\{S_{\\eta}\\}_{\\eta \\in [1,2]}$, called symmetric doubling maps, defined on $[-1,1]$ by $S_\\eta (x)=2x-d\\eta$, where $d\\in \\{-1,0,1 \\}$. Each map $S_\\eta$ generates binary expansions with digits $-1$, 0 and 1. We study the frequency of the digit 0 in typical expansions as a function of the parameter $\\eta$. The transformations $S_\\eta$ have a natural ergodic invariant measure $\\mu_\\eta$ that is absolutely continuous with respect to Lebesgue measure. The frequency of the digit 0 is related to the measure $\\mu_{\\eta}([-\\frac12,\\frac12])$ by the Ergodic "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.06335","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"}