{"paper":{"title":"Dirichlet improvability in $L_p$-norms","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.DS"],"primary_cat":"math.NT","authors_text":"Nikita Shulga, Nikolay Moshchevitin","submitted_at":"2024-08-12T14:49:47Z","abstract_excerpt":"For a norm $F$ on $\\mathbb{R}^2$, we consider the set of $F$-Dirichlet improvable numbers $\\mathbf{DI}_F$. In the most important case of $F$ being an $L_p$-norm with $p=\\infty$, which is a supremum norm, it is well-known that $\\mathbf{DI}_F = \\mathbf{BA}\\cup \\mathbb{Q}$, where $\\mathbf{BA}$ is a set of badly approximable numbers. It is also known that $\\mathbf{BA}$ and each $\\mathbf{DI}_F$ are of measure zero and of full Hausdorff dimension.\n  Using classification of critical lattices for unit balls in $L_p$, we provide a complete and effective characterization of $\\mathbf{DI}_p:=\\mathbf{DI}_{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.06200","kind":"arxiv","version":3},"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/2408.06200/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"}