{"paper":{"title":"Quasicircles of dimension 1+k^2 do not exist","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CV"],"primary_cat":"math.DS","authors_text":"Oleg Ivrii","submitted_at":"2015-11-23T14:36:11Z","abstract_excerpt":"A well-known theorem of S. Smirnov states that the Hausdorff dimension of a $k$-quasicircle is at most $1+k^2$. Here, we show that the precise upper bound $D(k) = 1+\\Sigma^2 k^2 + \\mathcal O(k^{8/3-\\varepsilon})$ where $\\Sigma^2$ is the maximal asymptotic variance of the Beurling transform, taken over the unit ball of $L^\\infty$. The quantity $\\Sigma^2$ was introduced in a joint work with K. Astala, A. Per\\\"al\\\"a and I. Prause where it was proved that $0.879 < \\Sigma^2 \\le 1$, while recently, H. Hedenmalm discovered that surprisingly $\\Sigma^2 <1$. We deduce the asymptotic expansion of $D(k)$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.07240","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"}