{"paper":{"title":"Approximately Jumping Towards the Origin","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.PR","authors_text":"Alex Albors, Braeden Sodt, Ding Yifan, Fran\\c{c}ois Cl\\'ement, Shosuke Kiami, Tony Zeng","submitted_at":"2024-12-05T16:04:32Z","abstract_excerpt":"Given an initial point $x_0 \\in \\mathbb{R}^d$ and a sequence of vectors $v_1, v_2, \\dots$ in $\\mathbb{R}^d$, we define a greedy sequence by setting $x_{n} = x_{n-1} \\pm v_n$ where the sign is chosen so as to minimize $\\|x_n\\|$. We prove that if the vectors $v_i$ are chosen uniformly at random from $\\mathbb{S}^{d-1}$ then elements of the sequence are, on average, approximately at distance $\\|x_n\\| \\sim \\sqrt{\\pi d/8}$ from the origin. We show that the sequence $(\\|x_n\\|)_{n=1}^{\\infty}$ has an invariant measure $\\pi_d$ depending only on $d$ and we determine its mean and study its decay for all "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.04284","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/2412.04284/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"}