{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:2SOAHHPMNGETS6YOJSEPJKGC2W","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":"e55e677ac9f08d706b474d1967b36ebdcc253b10dfd0acff959d61afddd7f272","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-09T20:37:57Z","title_canon_sha256":"7e8511d3048981005366a71c833d1e3bf36e3cd9febe1642eaf883cfd2c54a2c"},"schema_version":"1.0","source":{"id":"1311.2209","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.2209","created_at":"2026-05-18T01:15:59Z"},{"alias_kind":"arxiv_version","alias_value":"1311.2209v1","created_at":"2026-05-18T01:15:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.2209","created_at":"2026-05-18T01:15:59Z"},{"alias_kind":"pith_short_12","alias_value":"2SOAHHPMNGET","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"2SOAHHPMNGETS6YO","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"2SOAHHPM","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:606b8abef5aa4cb1de0cd27c96ec61c408c4211cf68988971447b6776866229d","target":"graph","created_at":"2026-05-18T01:15:59Z","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":"Let $Q$ be a fundamental domain of some full-rank lattice in ${\\Bbb R}^d$ and let $\\mu$ and $\\nu$ be two positive Borel measures on ${\\Bbb R}^d$ such that the convolution $\\mu\\ast\\nu$ is a multiple of $\\chi_Q$. We consider the problem as to whether or not both measures must be spectral (i.e. each of their respective associated $L^2$ space admits an orthogonal basis of exponentials) and we show that this is the case when $Q = [0,1]^d$. This theorem yields a large class of examples of spectral measures which are either absolutely continuous, singularly continuous or purely discrete spectral meas","authors_text":"Chun-kit Lai, Jean-Pierre Gabardo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-09T20:37:57Z","title":"Spectral measures associated with the factorization of the Lebesgue measure on a set via convolution"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.2209","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:d03397fff3d886144e46d98e8cbb10009c380b45b5f32e36e392266e7aabf06f","target":"record","created_at":"2026-05-18T01:15:59Z","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":"e55e677ac9f08d706b474d1967b36ebdcc253b10dfd0acff959d61afddd7f272","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.FA","submitted_at":"2013-11-09T20:37:57Z","title_canon_sha256":"7e8511d3048981005366a71c833d1e3bf36e3cd9febe1642eaf883cfd2c54a2c"},"schema_version":"1.0","source":{"id":"1311.2209","kind":"arxiv","version":1}},"canonical_sha256":"d49c039dec6989397b0e4c88f4a8c2d5823ef3de26ba56ce0b5f253bd3f8e9d1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d49c039dec6989397b0e4c88f4a8c2d5823ef3de26ba56ce0b5f253bd3f8e9d1","first_computed_at":"2026-05-18T01:15:59.284796Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:15:59.284796Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"W6igw0MyJTmpb5P3Yjr8iyNkYxELbA5h0tehruRwzlQ0vlBisOX1uThufdumAWkk2h20V0JZUMQKrJvu/wm0Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:15:59.285538Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.2209","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d03397fff3d886144e46d98e8cbb10009c380b45b5f32e36e392266e7aabf06f","sha256:606b8abef5aa4cb1de0cd27c96ec61c408c4211cf68988971447b6776866229d"],"state_sha256":"b2e498b9260ed157a3d29956fe92586e613a7cf036962413f572bba77ccbd1bf"}