{"paper":{"title":"On density of infinite subsets I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Changguang Dong","submitted_at":"2017-09-17T01:33:12Z","abstract_excerpt":"Let $Y$ be a compact metric space, $G$ be a group acting by transformations on $Y$. For any infinite subset $A\\subset Y$, we study the density of $gA$ for $g\\in G$ and quantitative density of the set $\\displaystyle{\\bigcup_{g\\in G_n}gA}$ by the Hausdorff semimetric $d^H$. It is proven that for any integer $n\\ge 2$, $\\epsilon>0$, any infinite subset $A\\subset \\mathbb T^n$, there is a $g\\in SL(n,\\mathbb Z)$ such that $gA$ is $\\epsilon$-dense. We also show that, for any infinite subset $A\\subset [0,1]$, for generic rotation and generic 3-IET, $$\\liminf_nn\\cdot d^H\\left(\\bigcup_{k=0}^{n-1}T^kA,[0,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.05591","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":""},"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"}