{"paper":{"title":"A curious dynamical system in the plane","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Stefan Steinerberger, Tony Zeng","submitted_at":"2024-09-13T16:26:19Z","abstract_excerpt":"For any irrational $\\alpha > 0$ and any initial value $z_{-1} \\in \\mathbb{C}$, we define a sequence of complex numbers $(z_n)_{n=0}^{\\infty}$ as follows: $z_n$ is $z_{n-1} + e^{2 \\pi i \\alpha n}$ or $z_{n-1} - e^{2 \\pi i \\alpha n}$, whichever has the smaller absolute value. If both numbers have the same absolute value, the sequence terminates at $z_{n-1}$ but this happens rarely. This dynamical system has astonishingly intricate behavior: the choice of signs in $z_{n-1} \\pm e^{2 \\pi i \\alpha n}$ appears to eventually become periodic (though the period can be large). We prove that if one observ"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.08961","kind":"arxiv","version":2},"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/2409.08961/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"}