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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}_{"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2408.06200","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-08-12T14:49:47Z","cross_cats_sorted":["math.DS"],"title_canon_sha256":"ba94efbe804743e920a368b089bba4bfd410ae35c944a3f01b2faa0962ef3d0a","abstract_canon_sha256":"7dc7e70baeaa627955b597c64230e809aed2343d96bc7af5447ead4b3ae2a388"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-23T03:13:43.633340Z","signature_b64":"8fj6F629MluPox9JLMQEOCvBQf9NcfJcp8Pge/5KEt+U4R5MiCu0G2pz3vlTtgPi91Q/LZy5AJinkSQ/bGzSDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"08c06906246fc4cb8ccf6f9f95731cbabd5fcf72ac4f0848600b520f1497d08d","last_reissued_at":"2026-06-23T03:13:43.632888Z","signature_status":"signed_v1","first_computed_at":"2026-06-23T03:13:43.632888Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathbf{DI}_F$. 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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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2408.06200","created_at":"2026-06-23T03:13:43.632944+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.06200v3","created_at":"2026-06-23T03:13:43.632944+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.06200","created_at":"2026-06-23T03:13:43.632944+00:00"},{"alias_kind":"pith_short_12","alias_value":"BDAGSBREN7CM","created_at":"2026-06-23T03:13:43.632944+00:00"},{"alias_kind":"pith_short_16","alias_value":"BDAGSBREN7CMXDGP","created_at":"2026-06-23T03:13:43.632944+00:00"},{"alias_kind":"pith_short_8","alias_value":"BDAGSBRE","created_at":"2026-06-23T03:13:43.632944+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2606.08865","citing_title":"The Dirichlet spectrum with respect to $L_1$ norm is $\\left[\\frac12,1\\right]$","ref_index":23,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK","json":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK.json","graph_json":"https://pith.science/api/pith-number/BDAGSBREN7CMXDGPN6PZK4Y4XK/graph.json","events_json":"https://pith.science/api/pith-number/BDAGSBREN7CMXDGPN6PZK4Y4XK/events.json","paper":"https://pith.science/paper/BDAGSBRE"},"agent_actions":{"view_html":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK","download_json":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK.json","view_paper":"https://pith.science/paper/BDAGSBRE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.06200&json=true","fetch_graph":"https://pith.science/api/pith-number/BDAGSBREN7CMXDGPN6PZK4Y4XK/graph.json","fetch_events":"https://pith.science/api/pith-number/BDAGSBREN7CMXDGPN6PZK4Y4XK/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK/action/storage_attestation","attest_author":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK/action/author_attestation","sign_citation":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK/action/citation_signature","submit_replication":"https://pith.science/pith/BDAGSBREN7CMXDGPN6PZK4Y4XK/action/replication_record"}},"created_at":"2026-06-23T03:13:43.632944+00:00","updated_at":"2026-06-23T03:13:43.632944+00:00"}