{"paper":{"title":"Quantitative recurrence properties and homogeneous self-similar sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.DS","authors_text":"Min Wu, Wen Wu, Yuanyang Chang","submitted_at":"2018-01-31T08:04:21Z","abstract_excerpt":"Let $K$ be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map $T: K\\rightarrow K$ induced by the shift. Let $\\mu$ be the natural self-similar measure supported on $K$. For a positive function $\\varphi$ defined on $\\mathbb{N}$, we show that the $\\mu$-measure of the following set \\begin{equation*}\n  R(\\varphi):=\\{x\\in K: |T^n x-x|<\\varphi(n) \\; \\text{for infinitely many} \\; n\\in\\mathbb{N}\\} \\end{equation*} is null or full according to convergence or divergence of a certain series. Moreo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.10334","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"}