{"paper":{"title":"On a Conjecture of Cusick on a sum of Cantor sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Nikita Shulga","submitted_at":"2024-11-26T12:39:28Z","abstract_excerpt":"In 1971 Cusick proved that every real number $x\\in[0,1]$ can be expressed as a sum of two continued fractions with no partial quotients equal to $1$. In other words, if we define a set $$ S(k):= \\{ x\\in[0,1] : a_n(x) \\geq k \\text{ for all } n\\in\\mathbb{N} \\}, $$ then $$ S(2)+S(2) = [0,1]. $$ He also conjectured that this result is unique in the sense that if you exclude partial quotients from $1$ to $k-1$ with $k\\geq3$, then the Lebesgue measure $\\lambda$ of the set of numbers which can be expressed as a sum of two continued fractions with no partial quotients from $\\{1,\\ldots,k-1\\}$ is equal "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2411.17379","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2411.17379/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}