{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:ETC7JOANDAEKBAPVHAZUQI636Z","short_pith_number":"pith:ETC7JOAN","schema_version":"1.0","canonical_sha256":"24c5f4b80d1808a081f538334823dbf65127539331228ad58ad9a33ff19c9672","source":{"kind":"arxiv","id":"1606.07596","version":4},"attestation_state":"computed","paper":{"title":"Spherical Recurrence and locally isometric embeddings of trees into positive density subsets of $\\mathbb{Z}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Kamil Bulinski","submitted_at":"2016-06-24T08:15:43Z","abstract_excerpt":"Magyar has shown that if $B \\subset \\mathbb{Z}^d$ has positive upper density $(d \\geq 5)$, then the set of squared distances $\\{ \\|b_1-b_2 \\|^2 \\text{ }: \\text{ } b_1,b_2 \\in B \\}$ contains an infinitely long arithmetic progression, whose period depends only on the upper density of $B$. We extend this result by showing that $B$ contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of $B$ and the number of vertices of the sought tree). In particular, $B$ contains all chains of elements "},"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":"1606.07596","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2016-06-24T08:15:43Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"006fab853e77d9661b58e7a28303411eabde5e1a83cdf305fc37700cbfcd96f0","abstract_canon_sha256":"a65fe706f170948389d466954d79d3516cb19c2b29612c87133d78df2dae9dc3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:42:22.005752Z","signature_b64":"IjFECgfqHVVBVyc9N1OL53M4iTSQe9+fuVmcto9l77fvrjXepG1I7oEbQyWSwGB2pTF87OpRaVcn9kMn9E/eAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"24c5f4b80d1808a081f538334823dbf65127539331228ad58ad9a33ff19c9672","last_reissued_at":"2026-05-18T00:42:22.005248Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:42:22.005248Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spherical Recurrence and locally isometric embeddings of trees into positive density subsets of $\\mathbb{Z}^d$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.DS","authors_text":"Kamil Bulinski","submitted_at":"2016-06-24T08:15:43Z","abstract_excerpt":"Magyar has shown that if $B \\subset \\mathbb{Z}^d$ has positive upper density $(d \\geq 5)$, then the set of squared distances $\\{ \\|b_1-b_2 \\|^2 \\text{ }: \\text{ } b_1,b_2 \\in B \\}$ contains an infinitely long arithmetic progression, whose period depends only on the upper density of $B$. We extend this result by showing that $B$ contains locally isometrically embedded copies of every tree with edge lengths in some given arithmetic progression (whose period depends only on the upper density of $B$ and the number of vertices of the sought tree). In particular, $B$ contains all chains of elements "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.07596","kind":"arxiv","version":4},"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":"1606.07596","created_at":"2026-05-18T00:42:22.005334+00:00"},{"alias_kind":"arxiv_version","alias_value":"1606.07596v4","created_at":"2026-05-18T00:42:22.005334+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.07596","created_at":"2026-05-18T00:42:22.005334+00:00"},{"alias_kind":"pith_short_12","alias_value":"ETC7JOANDAEK","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_16","alias_value":"ETC7JOANDAEKBAPV","created_at":"2026-05-18T12:30:15.759754+00:00"},{"alias_kind":"pith_short_8","alias_value":"ETC7JOAN","created_at":"2026-05-18T12:30:15.759754+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/ETC7JOANDAEKBAPVHAZUQI636Z","json":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z.json","graph_json":"https://pith.science/api/pith-number/ETC7JOANDAEKBAPVHAZUQI636Z/graph.json","events_json":"https://pith.science/api/pith-number/ETC7JOANDAEKBAPVHAZUQI636Z/events.json","paper":"https://pith.science/paper/ETC7JOAN"},"agent_actions":{"view_html":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z","download_json":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z.json","view_paper":"https://pith.science/paper/ETC7JOAN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1606.07596&json=true","fetch_graph":"https://pith.science/api/pith-number/ETC7JOANDAEKBAPVHAZUQI636Z/graph.json","fetch_events":"https://pith.science/api/pith-number/ETC7JOANDAEKBAPVHAZUQI636Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z/action/storage_attestation","attest_author":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z/action/author_attestation","sign_citation":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z/action/citation_signature","submit_replication":"https://pith.science/pith/ETC7JOANDAEKBAPVHAZUQI636Z/action/replication_record"}},"created_at":"2026-05-18T00:42:22.005334+00:00","updated_at":"2026-05-18T00:42:22.005334+00:00"}