{"paper":{"title":"Multifractal analysis of the Birkhoff sums of Saint-Petersburg potential","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Baowei Wang, Dong Han Kim, Lingmin Liao (LAMA), Michal Rams (PAN)","submitted_at":"2017-07-19T13:00:40Z","abstract_excerpt":"Let $((0,1], T)$ be the doubling map in the unit interval and $\\varphi$ be the Saint-Petersburg potential, defined by $\\varphi(x)=2^n$ if $x\\in (2^{-n-1}, 2^{-n}]$ for all $n\\geq 0$. We consider the asymptotic properties of the Birkhoff sum $S\\_n(x)=\\varphi(x)+\\cdots+\\varphi(T^{n-1}(x))$. With respect to the Lebesgue measure, the Saint-Petersburg potential is not integrable and it is known that $\\frac{1}{n\\log n}S\\_n(x)$ converges to $\\frac{1}{\\log 2}$ in probability. We determine the Hausdorff dimension of the level set $\\{x: \\lim\\_{n\\to\\infty}S\\_n(x)/n=\\alpha\\} \\ (\\alpha>0)$, as well as that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06059","kind":"arxiv","version":2},"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"}