{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:XVZKZWWQYR3L5HI5CKH5NOLQNN","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f8217204d16c99d194fad9a50b9faba21d25c3405e95c2504a8b65d7fa08008b","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-05-05T05:24:44Z","title_canon_sha256":"c1864ca0b33a9a3e7698956103331e6b324bcca5cc7b47eeb347ed06eb25dcff"},"schema_version":"1.0","source":{"id":"1505.00883","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1505.00883","created_at":"2026-05-18T00:42:33Z"},{"alias_kind":"arxiv_version","alias_value":"1505.00883v1","created_at":"2026-05-18T00:42:33Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1505.00883","created_at":"2026-05-18T00:42:33Z"},{"alias_kind":"pith_short_12","alias_value":"XVZKZWWQYR3L","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_16","alias_value":"XVZKZWWQYR3L5HI5","created_at":"2026-05-18T12:29:50Z"},{"alias_kind":"pith_short_8","alias_value":"XVZKZWWQ","created_at":"2026-05-18T12:29:50Z"}],"graph_snapshots":[{"event_id":"sha256:4bdeb3812eb0c772a519779a20c7012949f8c6f4205a0323253cc8a212bf050b","target":"graph","created_at":"2026-05-18T00:42:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we study subsets $E$ of ${\\Bbb Z}_p^d$ such that any function $f: E \\to {\\Bbb C}$ can be written as a linear combination of characters orthogonal with respect to $E$. We shall refer to such sets as spectral. In this context, we prove the Fuglede Conjecture in ${\\Bbb Z}_p^2$ which says that $E \\subset {\\Bbb Z}_p^2$ is spectral if and only if $E$ tiles ${\\Bbb Z}_p^2$ by translation. Arithmetic properties of the finite field Fourier transform, elementary Galois theory and combinatorial geometric properties of direction sets play the key role in the proof.","authors_text":"Alex Iosevich, Azita Mayeli, Jonathan Pakianathan","cross_cats":["math.CO","math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-05-05T05:24:44Z","title":"The Fuglede Conjecture holds in ${\\Bbb Z}_p \\times {\\Bbb Z}_p$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.00883","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:870cbf2c7b7e36aa33892d13db923821d7eb4704bf20549290a4619986541d53","target":"record","created_at":"2026-05-18T00:42:33Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f8217204d16c99d194fad9a50b9faba21d25c3405e95c2504a8b65d7fa08008b","cross_cats_sorted":["math.CO","math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2015-05-05T05:24:44Z","title_canon_sha256":"c1864ca0b33a9a3e7698956103331e6b324bcca5cc7b47eeb347ed06eb25dcff"},"schema_version":"1.0","source":{"id":"1505.00883","kind":"arxiv","version":1}},"canonical_sha256":"bd72acdad0c476be9d1d128fd6b9706b79be069378743f44174cacfd6bdf9000","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bd72acdad0c476be9d1d128fd6b9706b79be069378743f44174cacfd6bdf9000","first_computed_at":"2026-05-18T00:42:33.488610Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:33.488610Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"UMG8TLWwTGZFPJikO2ee+xJOrNGTuC8wAfKnVLqjHe0ruy2sLiCUZqie4WJeW9nMthmyG/BEwBwfwDRJB4gPCA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:33.489230Z","signed_message":"canonical_sha256_bytes"},"source_id":"1505.00883","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:870cbf2c7b7e36aa33892d13db923821d7eb4704bf20549290a4619986541d53","sha256:4bdeb3812eb0c772a519779a20c7012949f8c6f4205a0323253cc8a212bf050b"],"state_sha256":"29feff2ae9b80d77a1e8f7226b361c0036e02f8f251becfc483ac0f61f16dc29"}