{"paper":{"title":"Strong marker sets and applications","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.LO","authors_text":"Su Gao, Tianhao Wang","submitted_at":"2025-02-01T23:53:52Z","abstract_excerpt":"We prove the existence of clopen marker sets with some strong regularity property. For each $n\\geq 1$ and any integer $d\\geq 1$, we show that there are a positive integer $D$ and a clopen marker set $M$ in $F(2^{\\mathbb{Z}^n})$ such that (1) for any distinct $x,y\\in M$ in the same orbit, $\\rho(x,y)\\geq d$; (2) for any $1\\leq i\\leq n$ and any $x\\in F(2^{\\mathbb{Z}^n})$, there are non-negative integers $a, b\\leq D$ such that $a\\cdot x\\in M$ and $-b\\cdot x\\in M$. As an application, we obtain a clopen tree section for $F(2^{\\mathbb{Z}^n})$. Based on the strong marker sets, we get a quick proof tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.00598","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/2502.00598/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"}