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We prove that for any $n$-dimensional metric compactum $X$ each of the sets $\\mathcal H(3,1,m,3n+1-m)$ and $\\mathcal H(2,1,m,2n)$ is dense and $G_\\delta$ in the function space $C(X,\\mathbb R^m)$ provided $m\\geq 2n+1$ (in this case $\\mathcal H(3,1,m,3n+1-m)$ and $\\mathcal H(2,1,m,2n)$ can consist of embeddings)"},"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":"1010.1892","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GN","submitted_at":"2010-10-10T03:59:17Z","cross_cats_sorted":[],"title_canon_sha256":"adfa8cd30f49511c81f8d3e9bcd0e3822e5673337609a3cd8b95b65eedcc220f","abstract_canon_sha256":"a7ed5a7047f3e9ac70b143616dab3b65daff8af7a8e31a5a9f6a5c4f0291b9aa"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:36:51.062264Z","signature_b64":"kt0OnePeyeWnv3xHltt0ebtqTTRJdYm8kK6O55kf8VZokSj0Yprf4jzEMd28842D1XwV9PR0kw6v0cjnm58EBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a714d49064d9548c88c24ab4c2dfc736295e2b476a7d17ebbfac2e5194188c63","last_reissued_at":"2026-05-18T04:36:51.061837Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:36:51.061837Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Embeddings of finite-dimensional compacta in Euclidean spaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GN","authors_text":"S. Bogataya, S. Bogatyi, V. Valov","submitted_at":"2010-10-10T03:59:17Z","abstract_excerpt":"If $g$ is a map from a space $X$ into $\\mathbb R^m$ and $q$ is an integer, let $B_{q,d,m}(g)$ be the set of all lines $\\Pi^d\\subset\\mathbb R^m$ such that $|g^{-1}(\\Pi^d)|\\geq q$. Let also $\\mathcal H(q,d,m,k)$ denote the maps $g\\colon X\\to\\mathbb R^m$ such that $\\dim B_{q,d,m}(g)\\leq k$. We prove that for any $n$-dimensional metric compactum $X$ each of the sets $\\mathcal H(3,1,m,3n+1-m)$ and $\\mathcal H(2,1,m,2n)$ is dense and $G_\\delta$ in the function space $C(X,\\mathbb R^m)$ provided $m\\geq 2n+1$ (in this case $\\mathcal H(3,1,m,3n+1-m)$ and $\\mathcal H(2,1,m,2n)$ can consist of embeddings)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1010.1892","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1010.1892","created_at":"2026-05-18T04:36:51.061904+00:00"},{"alias_kind":"arxiv_version","alias_value":"1010.1892v2","created_at":"2026-05-18T04:36:51.061904+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1010.1892","created_at":"2026-05-18T04:36:51.061904+00:00"},{"alias_kind":"pith_short_12","alias_value":"U4KNJEDE3FKI","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_16","alias_value":"U4KNJEDE3FKIZCGC","created_at":"2026-05-18T12:26:15.391820+00:00"},{"alias_kind":"pith_short_8","alias_value":"U4KNJEDE","created_at":"2026-05-18T12:26:15.391820+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY","json":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY.json","graph_json":"https://pith.science/api/pith-number/U4KNJEDE3FKIZCGCJK2MFX6HGY/graph.json","events_json":"https://pith.science/api/pith-number/U4KNJEDE3FKIZCGCJK2MFX6HGY/events.json","paper":"https://pith.science/paper/U4KNJEDE"},"agent_actions":{"view_html":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY","download_json":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY.json","view_paper":"https://pith.science/paper/U4KNJEDE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1010.1892&json=true","fetch_graph":"https://pith.science/api/pith-number/U4KNJEDE3FKIZCGCJK2MFX6HGY/graph.json","fetch_events":"https://pith.science/api/pith-number/U4KNJEDE3FKIZCGCJK2MFX6HGY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY/action/storage_attestation","attest_author":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY/action/author_attestation","sign_citation":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY/action/citation_signature","submit_replication":"https://pith.science/pith/U4KNJEDE3FKIZCGCJK2MFX6HGY/action/replication_record"}},"created_at":"2026-05-18T04:36:51.061904+00:00","updated_at":"2026-05-18T04:36:51.061904+00:00"}