{"paper":{"title":"On exceptional sets in Erd\\H{o}s-R\\'{e}nyi limit theorem revisited","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Jinjun Li, Min Wu","submitted_at":"2015-11-28T15:06:04Z","abstract_excerpt":"For $x\\in [0,1],$ the run-length function $r_n(x)$ is defined as the length of the longest run of $1$'s amongst the first $n$ dyadic digits in the dyadic expansion of $x.$ Erd\\H{o}s and R\\'enyi proved that $\\lim\\limits_{n\\to\\infty}\\frac{r_n(x)}{\\log_2n}=1$ for Lebesgue almost all $x\\in[0,1]$. Let $H$ denote the set of monotonically increasing functions $\\varphi:\\mathbb{N}\\to (0,+\\infty)$ with $\\lim\\limits_{n\\to\\infty}\\varphi(n)=+\\infty$. For any $\\varphi\\in H$, we prove that the set \\[ E_{\\max}^\\varphi=\\left\\{x\\in [0,1]:\\liminf\\limits_{n\\to\\infty}\\frac{r_n(x)}{\\varphi(n)}=0, \\limsup\\limits_{n\\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.08903","kind":"arxiv","version":4},"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"}