{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:NGCSSIMFE2C4LZ5WNJSI25V54W","short_pith_number":"pith:NGCSSIMF","canonical_record":{"source":{"id":"1807.02619","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-07-07T06:29:55Z","cross_cats_sorted":[],"title_canon_sha256":"69419e720a4ba74ba4c78dbf8089ac31e320c2f6678c8c5edb6c9c2898a49250","abstract_canon_sha256":"a896f8d2f1eb9de6d4e2d7c6a091af4258eea7befee01902426be3430a306165"},"schema_version":"1.0"},"canonical_sha256":"69852921852685c5e7b66a648d76bde5b16e1492bbd38af572753deec1544f18","source":{"kind":"arxiv","id":"1807.02619","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.02619","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1807.02619v3","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02619","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"NGCSSIMFE2C4","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NGCSSIMFE2C4LZ5W","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NGCSSIMF","created_at":"2026-05-18T12:32:40Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:NGCSSIMFE2C4LZ5WNJSI25V54W","target":"record","payload":{"canonical_record":{"source":{"id":"1807.02619","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-07-07T06:29:55Z","cross_cats_sorted":[],"title_canon_sha256":"69419e720a4ba74ba4c78dbf8089ac31e320c2f6678c8c5edb6c9c2898a49250","abstract_canon_sha256":"a896f8d2f1eb9de6d4e2d7c6a091af4258eea7befee01902426be3430a306165"},"schema_version":"1.0"},"canonical_sha256":"69852921852685c5e7b66a648d76bde5b16e1492bbd38af572753deec1544f18","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:41:01.247151Z","signature_b64":"xyUzFbF3E3qqcxqX6//ZnrYvE2m/8gqDK/VQNZmYAyfsndI3I8Cl7fmKrcYnHGp/C7Y76NGWnUAvwfDDIAIMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"69852921852685c5e7b66a648d76bde5b16e1492bbd38af572753deec1544f18","last_reissued_at":"2026-05-17T23:41:01.246408Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:41:01.246408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1807.02619","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Z/6XUGkPFCv+UsjRU4LXBpcQeduhFDhz2zLhyqeQY+XVM0OuRV56ABP/XBIGTCr83DoOHSApRU7Rd5eyi+pmCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T04:32:07.424901Z"},"content_sha256":"2e972709ebf9cd0fabe0f8dc4bc7c31eec1f067d25be3c0938f9feea8572ec9a","schema_version":"1.0","event_id":"sha256:2e972709ebf9cd0fabe0f8dc4bc7c31eec1f067d25be3c0938f9feea8572ec9a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:NGCSSIMFE2C4LZ5WNJSI25V54W","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On syndetic Riesz sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Itay Londner, Marcin Bownik","submitted_at":"2018-07-07T06:29:55Z","abstract_excerpt":"Applying the solution to the Kadison-Singer problem, we show that every subset $\\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\\left\\{ e^{i\\lambda x}\\right\\} _{\\lambda \\in \\Lambda}$ such that $\\Lambda\\subset\\mathbb{Z}$ is a set with gaps between consecutive elements bounded by ${\\displaystyle \\frac{C}{\\left|\\mathcal{S}\\right|}}$. In the case when $\\mathcal{S}$ is an open set we demonstrate, using quasicrystals, how such $\\Lambda$ can be deterministically constructed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02619","kind":"arxiv","version":3},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:41:01Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SvNudYCdIt9pOSArUVS42KuTmv9TIO8RS6WP4pNCGNrjQAL11Bw2xSGiD5OqcELtHkGDF8rbBuEh02xNeEHCCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-26T04:32:07.425249Z"},"content_sha256":"4d8ff8ebdacef452ce83689ea96b5314786ff0129a2ba61f5a73db96535fc5d2","schema_version":"1.0","event_id":"sha256:4d8ff8ebdacef452ce83689ea96b5314786ff0129a2ba61f5a73db96535fc5d2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NGCSSIMFE2C4LZ5WNJSI25V54W/bundle.json","state_url":"https://pith.science/pith/NGCSSIMFE2C4LZ5WNJSI25V54W/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NGCSSIMFE2C4LZ5WNJSI25V54W/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-26T04:32:07Z","links":{"resolver":"https://pith.science/pith/NGCSSIMFE2C4LZ5WNJSI25V54W","bundle":"https://pith.science/pith/NGCSSIMFE2C4LZ5WNJSI25V54W/bundle.json","state":"https://pith.science/pith/NGCSSIMFE2C4LZ5WNJSI25V54W/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NGCSSIMFE2C4LZ5WNJSI25V54W/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:NGCSSIMFE2C4LZ5WNJSI25V54W","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":"a896f8d2f1eb9de6d4e2d7c6a091af4258eea7befee01902426be3430a306165","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-07-07T06:29:55Z","title_canon_sha256":"69419e720a4ba74ba4c78dbf8089ac31e320c2f6678c8c5edb6c9c2898a49250"},"schema_version":"1.0","source":{"id":"1807.02619","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1807.02619","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"arxiv_version","alias_value":"1807.02619v3","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1807.02619","created_at":"2026-05-17T23:41:01Z"},{"alias_kind":"pith_short_12","alias_value":"NGCSSIMFE2C4","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_16","alias_value":"NGCSSIMFE2C4LZ5W","created_at":"2026-05-18T12:32:40Z"},{"alias_kind":"pith_short_8","alias_value":"NGCSSIMF","created_at":"2026-05-18T12:32:40Z"}],"graph_snapshots":[{"event_id":"sha256:4d8ff8ebdacef452ce83689ea96b5314786ff0129a2ba61f5a73db96535fc5d2","target":"graph","created_at":"2026-05-17T23:41:01Z","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":"Applying the solution to the Kadison-Singer problem, we show that every subset $\\mathcal{S}$ of the torus of positive Lebesgue measure admits a Riesz sequence of exponentials $\\left\\{ e^{i\\lambda x}\\right\\} _{\\lambda \\in \\Lambda}$ such that $\\Lambda\\subset\\mathbb{Z}$ is a set with gaps between consecutive elements bounded by ${\\displaystyle \\frac{C}{\\left|\\mathcal{S}\\right|}}$. In the case when $\\mathcal{S}$ is an open set we demonstrate, using quasicrystals, how such $\\Lambda$ can be deterministically constructed.","authors_text":"Itay Londner, Marcin Bownik","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-07-07T06:29:55Z","title":"On syndetic Riesz sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1807.02619","kind":"arxiv","version":3},"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":"sha256:2e972709ebf9cd0fabe0f8dc4bc7c31eec1f067d25be3c0938f9feea8572ec9a","target":"record","created_at":"2026-05-17T23:41:01Z","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":"a896f8d2f1eb9de6d4e2d7c6a091af4258eea7befee01902426be3430a306165","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-07-07T06:29:55Z","title_canon_sha256":"69419e720a4ba74ba4c78dbf8089ac31e320c2f6678c8c5edb6c9c2898a49250"},"schema_version":"1.0","source":{"id":"1807.02619","kind":"arxiv","version":3}},"canonical_sha256":"69852921852685c5e7b66a648d76bde5b16e1492bbd38af572753deec1544f18","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"69852921852685c5e7b66a648d76bde5b16e1492bbd38af572753deec1544f18","first_computed_at":"2026-05-17T23:41:01.246408Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:41:01.246408Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xyUzFbF3E3qqcxqX6//ZnrYvE2m/8gqDK/VQNZmYAyfsndI3I8Cl7fmKrcYnHGp/C7Y76NGWnUAvwfDDIAIMCA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:41:01.247151Z","signed_message":"canonical_sha256_bytes"},"source_id":"1807.02619","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2e972709ebf9cd0fabe0f8dc4bc7c31eec1f067d25be3c0938f9feea8572ec9a","sha256:4d8ff8ebdacef452ce83689ea96b5314786ff0129a2ba61f5a73db96535fc5d2"],"state_sha256":"6271964eaa1292e2a352488467bfdc2f086d1b53eefb71b597cac663127eef5d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"LKs+7k4zZ4CIWCTbMsBQ9qRf7lf4/8zGcSRTgVg5/oMWR5ctHsDvD+xyVjprBF0Fy9hcjuO13GVseyUgkcJgDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-26T04:32:07.427296Z","bundle_sha256":"eeb587df9e90d046a8de1357782d066dd629fd10da4a937254c82b19ea2b92ff"}}