{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:U6QHN6NGAQYJ2BOGF2WMJ6GV26","short_pith_number":"pith:U6QHN6NG","schema_version":"1.0","canonical_sha256":"a7a076f9a604309d05c62eacc4f8d5d78b7689e8a7006501f88716a9bcd29f62","source":{"kind":"arxiv","id":"1503.06639","version":4},"attestation_state":"computed","paper":{"title":"A finite version of the Kakeya problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aart Blokhuis, Diego Domenzain, Simeon Ball","submitted_at":"2015-03-23T14:04:41Z","abstract_excerpt":"Let $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered as points of the projective space at infinity. We give a geometric construction of a set of lines $L$, where $D$ contains an $N^{n-1}$ grid and where $S$ has size $2((1/2)N)^n$, given a starting configuration in the plane. We provide examples of such starting configurations for the reals and for finite fields. Following Dvir's proof of the finite field Kakeya"},"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":"1503.06639","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-03-23T14:04:41Z","cross_cats_sorted":[],"title_canon_sha256":"12523de870abbd009e64f4b1ca4b9274e0f904de2c5df2758c0f132602da0667","abstract_canon_sha256":"920c8a3caf2af24f77c22718945c43b1d356a66eb93aa50b782e3b1ee61f7d3d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:15:53.048529Z","signature_b64":"f6TWpUo7hj/I8LXNfN1az68UxNtzOe0jXTELV9tz0/fJADTRasCTCfLiW+0Jwufco6/JBAa6+NCSUBRXfwZJBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a7a076f9a604309d05c62eacc4f8d5d78b7689e8a7006501f88716a9bcd29f62","last_reissued_at":"2026-05-18T01:15:53.048093Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:15:53.048093Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A finite version of the Kakeya problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Aart Blokhuis, Diego Domenzain, Simeon Ball","submitted_at":"2015-03-23T14:04:41Z","abstract_excerpt":"Let $L$ be a set of lines of an affine space over a field and let $S$ be a set of points with the property that every line of $L$ is incident with at least $N$ points of $S$. Let $D$ be the set of directions of the lines of $L$ considered as points of the projective space at infinity. We give a geometric construction of a set of lines $L$, where $D$ contains an $N^{n-1}$ grid and where $S$ has size $2((1/2)N)^n$, given a starting configuration in the plane. We provide examples of such starting configurations for the reals and for finite fields. Following Dvir's proof of the finite field Kakeya"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.06639","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":"1503.06639","created_at":"2026-05-18T01:15:53.048163+00:00"},{"alias_kind":"arxiv_version","alias_value":"1503.06639v4","created_at":"2026-05-18T01:15:53.048163+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.06639","created_at":"2026-05-18T01:15:53.048163+00:00"},{"alias_kind":"pith_short_12","alias_value":"U6QHN6NGAQYJ","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_16","alias_value":"U6QHN6NGAQYJ2BOG","created_at":"2026-05-18T12:29:44.643036+00:00"},{"alias_kind":"pith_short_8","alias_value":"U6QHN6NG","created_at":"2026-05-18T12:29:44.643036+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/U6QHN6NGAQYJ2BOGF2WMJ6GV26","json":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26.json","graph_json":"https://pith.science/api/pith-number/U6QHN6NGAQYJ2BOGF2WMJ6GV26/graph.json","events_json":"https://pith.science/api/pith-number/U6QHN6NGAQYJ2BOGF2WMJ6GV26/events.json","paper":"https://pith.science/paper/U6QHN6NG"},"agent_actions":{"view_html":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26","download_json":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26.json","view_paper":"https://pith.science/paper/U6QHN6NG","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1503.06639&json=true","fetch_graph":"https://pith.science/api/pith-number/U6QHN6NGAQYJ2BOGF2WMJ6GV26/graph.json","fetch_events":"https://pith.science/api/pith-number/U6QHN6NGAQYJ2BOGF2WMJ6GV26/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26/action/timestamp_anchor","attest_storage":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26/action/storage_attestation","attest_author":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26/action/author_attestation","sign_citation":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26/action/citation_signature","submit_replication":"https://pith.science/pith/U6QHN6NGAQYJ2BOGF2WMJ6GV26/action/replication_record"}},"created_at":"2026-05-18T01:15:53.048163+00:00","updated_at":"2026-05-18T01:15:53.048163+00:00"}