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We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\\mathcal P \\subset \\mathcal G(n,k)$ such that for all $P\\in \\mathcal P$, there exists a translation vector $a_P \\in \\mathbb{R}^n$ such that $\\dim(F\\cap (P + a_P)) \\geq s$. We show that given $k \\geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;"},"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":"2412.18193","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2024-12-24T06:01:17Z","cross_cats_sorted":["math.MG"],"title_canon_sha256":"87be102e844f0cefc295b11e91fe19aa1d9c74dbffbdd5b41d203e41f22fce6a","abstract_canon_sha256":"7bba8876caaf075f9d7ca3bfbd67713768b3462bceb0bd96a67b61db7d6cea0a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:30:29.570386Z","signature_b64":"Nhlk+We6tbYj1+LYXlL/FH1Qnru+0NoANjI6WhmEPOaWJQnEOR6cRXEEthUemscME0JM3JNAEkneYEPBrAohBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"251a763b2f51a5fa7d00fc9c7abb7311858a70e8d55190e60e7f8fe346c711de","last_reissued_at":"2026-07-05T10:30:29.569879Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:30:29.569879Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Spread Furstenberg Sets","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CA","authors_text":"Manik Dhar, Paige Bright","submitted_at":"2024-12-24T06:01:17Z","abstract_excerpt":"We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\\mathbb{R}^n$. In particular, let $F\\subset \\mathbb{R}^n$, $1\\leq k \\leq n-1$, $s\\in (0,k]$, and $t\\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\\mathcal P \\subset \\mathcal G(n,k)$ such that for all $P\\in \\mathcal P$, there exists a translation vector $a_P \\in \\mathbb{R}^n$ such that $\\dim(F\\cap (P + a_P)) \\geq s$. We show that given $k \\geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.18193","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/2412.18193/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":"2412.18193","created_at":"2026-07-05T10:30:29.569938+00:00"},{"alias_kind":"arxiv_version","alias_value":"2412.18193v2","created_at":"2026-07-05T10:30:29.569938+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.18193","created_at":"2026-07-05T10:30:29.569938+00:00"},{"alias_kind":"pith_short_12","alias_value":"EUNHMOZPKGS7","created_at":"2026-07-05T10:30:29.569938+00:00"},{"alias_kind":"pith_short_16","alias_value":"EUNHMOZPKGS7U7IA","created_at":"2026-07-05T10:30:29.569938+00:00"},{"alias_kind":"pith_short_8","alias_value":"EUNHMOZP","created_at":"2026-07-05T10:30:29.569938+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2508.18257","citing_title":"On the packing dimension of unions and extensions of $k$-planes","ref_index":3,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG","json":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG.json","graph_json":"https://pith.science/api/pith-number/EUNHMOZPKGS7U7IA7SOHVO3TCG/graph.json","events_json":"https://pith.science/api/pith-number/EUNHMOZPKGS7U7IA7SOHVO3TCG/events.json","paper":"https://pith.science/paper/EUNHMOZP"},"agent_actions":{"view_html":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG","download_json":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG.json","view_paper":"https://pith.science/paper/EUNHMOZP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2412.18193&json=true","fetch_graph":"https://pith.science/api/pith-number/EUNHMOZPKGS7U7IA7SOHVO3TCG/graph.json","fetch_events":"https://pith.science/api/pith-number/EUNHMOZPKGS7U7IA7SOHVO3TCG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG/action/storage_attestation","attest_author":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG/action/author_attestation","sign_citation":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG/action/citation_signature","submit_replication":"https://pith.science/pith/EUNHMOZPKGS7U7IA7SOHVO3TCG/action/replication_record"}},"created_at":"2026-07-05T10:30:29.569938+00:00","updated_at":"2026-07-05T10:30:29.569938+00:00"}