{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:LP2AKZIGYHAY6SXEUCK7JGKOFB","short_pith_number":"pith:LP2AKZIG","canonical_record":{"source":{"id":"1309.1506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-05T22:52:52Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"7e99e9ce169713744d694f8e36133fe5e5bc0e25b755819fbd5c1c93642e3d46","abstract_canon_sha256":"a693e7a544fcd1dc0ddb651c65020be90849dd911f2691c9f6d4317af1aa02a8"},"schema_version":"1.0"},"canonical_sha256":"5bf4056506c1c18f4ae4a095f4994e286363cce2ed63884ea842fd7f52a91337","source":{"kind":"arxiv","id":"1309.1506","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.1506","created_at":"2026-05-18T03:13:59Z"},{"alias_kind":"arxiv_version","alias_value":"1309.1506v1","created_at":"2026-05-18T03:13:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.1506","created_at":"2026-05-18T03:13:59Z"},{"alias_kind":"pith_short_12","alias_value":"LP2AKZIGYHAY","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LP2AKZIGYHAY6SXE","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LP2AKZIG","created_at":"2026-05-18T12:27:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:LP2AKZIGYHAY6SXEUCK7JGKOFB","target":"record","payload":{"canonical_record":{"source":{"id":"1309.1506","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-05T22:52:52Z","cross_cats_sorted":["math.CA"],"title_canon_sha256":"7e99e9ce169713744d694f8e36133fe5e5bc0e25b755819fbd5c1c93642e3d46","abstract_canon_sha256":"a693e7a544fcd1dc0ddb651c65020be90849dd911f2691c9f6d4317af1aa02a8"},"schema_version":"1.0"},"canonical_sha256":"5bf4056506c1c18f4ae4a095f4994e286363cce2ed63884ea842fd7f52a91337","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:13:59.299705Z","signature_b64":"5MYX7E53dwMs+5RQzQOt19wZrqwdMPqkkcq9KfRVxjrtng5QTtZiLYpB935TKHoFb3Ce7XsuoSBgOHzkcexnAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5bf4056506c1c18f4ae4a095f4994e286363cce2ed63884ea842fd7f52a91337","last_reissued_at":"2026-05-18T03:13:59.299165Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:13:59.299165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1309.1506","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:13:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"YHNjMdlj5C6LBIHWIUSaaOkaphIc8CAX4tszHzPwNkkA/7FNRahSNXm6zw5q6ob3Fh7WPirNezz6s1YEIVckAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T23:39:17.652467Z"},"content_sha256":"a248b779677462a44c3bfc879785c9c1f1f202fba384187ba47e9f0d0ae4f9c7","schema_version":"1.0","event_id":"sha256:a248b779677462a44c3bfc879785c9c1f1f202fba384187ba47e9f0d0ae4f9c7"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:LP2AKZIGYHAY6SXEUCK7JGKOFB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Sums of products of fractional parts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Jeffrey D. Vaaler, Thai Hoang Le","submitted_at":"2013-09-05T22:52:52Z","abstract_excerpt":"We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood conjecture in Diophantine approximation. We introduce a generalization of such counterexamples which we call strongly badly approximable matrices. And we prove a transference principle for strongly badly approximable matrices."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1506","kind":"arxiv","version":1},"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":"2026-05-18T03:13:59Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"z5cLu+u/2IEoHijYaG0m/z8s1XAw9J6uc0mXlpaQui2fo0opHt6HuQtFxKMFz/4Jh+nOs5r+PC66//e8atv4Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T23:39:17.653008Z"},"content_sha256":"59d531db8d6d2ba08146a86179ac2d9aec24624bb22fd59268dc65cf49a28e7e","schema_version":"1.0","event_id":"sha256:59d531db8d6d2ba08146a86179ac2d9aec24624bb22fd59268dc65cf49a28e7e"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/LP2AKZIGYHAY6SXEUCK7JGKOFB/bundle.json","state_url":"https://pith.science/pith/LP2AKZIGYHAY6SXEUCK7JGKOFB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/LP2AKZIGYHAY6SXEUCK7JGKOFB/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":"2026-05-28T23:39:17Z","links":{"resolver":"https://pith.science/pith/LP2AKZIGYHAY6SXEUCK7JGKOFB","bundle":"https://pith.science/pith/LP2AKZIGYHAY6SXEUCK7JGKOFB/bundle.json","state":"https://pith.science/pith/LP2AKZIGYHAY6SXEUCK7JGKOFB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/LP2AKZIGYHAY6SXEUCK7JGKOFB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:LP2AKZIGYHAY6SXEUCK7JGKOFB","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":"a693e7a544fcd1dc0ddb651c65020be90849dd911f2691c9f6d4317af1aa02a8","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-05T22:52:52Z","title_canon_sha256":"7e99e9ce169713744d694f8e36133fe5e5bc0e25b755819fbd5c1c93642e3d46"},"schema_version":"1.0","source":{"id":"1309.1506","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1309.1506","created_at":"2026-05-18T03:13:59Z"},{"alias_kind":"arxiv_version","alias_value":"1309.1506v1","created_at":"2026-05-18T03:13:59Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1309.1506","created_at":"2026-05-18T03:13:59Z"},{"alias_kind":"pith_short_12","alias_value":"LP2AKZIGYHAY","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"LP2AKZIGYHAY6SXE","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"LP2AKZIG","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:59d531db8d6d2ba08146a86179ac2d9aec24624bb22fd59268dc65cf49a28e7e","target":"graph","created_at":"2026-05-18T03:13:59Z","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":"We prove upper and lower bounds for certain sums of products of fractional parts by using majoring and minorizing functions from Fourier analysis. In special cases the upper bounds are sharp if there exist counterexamples to the Littlewood conjecture in Diophantine approximation. We introduce a generalization of such counterexamples which we call strongly badly approximable matrices. And we prove a transference principle for strongly badly approximable matrices.","authors_text":"Jeffrey D. Vaaler, Thai Hoang Le","cross_cats":["math.CA"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-05T22:52:52Z","title":"Sums of products of fractional parts"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.1506","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:a248b779677462a44c3bfc879785c9c1f1f202fba384187ba47e9f0d0ae4f9c7","target":"record","created_at":"2026-05-18T03:13:59Z","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":"a693e7a544fcd1dc0ddb651c65020be90849dd911f2691c9f6d4317af1aa02a8","cross_cats_sorted":["math.CA"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-09-05T22:52:52Z","title_canon_sha256":"7e99e9ce169713744d694f8e36133fe5e5bc0e25b755819fbd5c1c93642e3d46"},"schema_version":"1.0","source":{"id":"1309.1506","kind":"arxiv","version":1}},"canonical_sha256":"5bf4056506c1c18f4ae4a095f4994e286363cce2ed63884ea842fd7f52a91337","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5bf4056506c1c18f4ae4a095f4994e286363cce2ed63884ea842fd7f52a91337","first_computed_at":"2026-05-18T03:13:59.299165Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:13:59.299165Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5MYX7E53dwMs+5RQzQOt19wZrqwdMPqkkcq9KfRVxjrtng5QTtZiLYpB935TKHoFb3Ce7XsuoSBgOHzkcexnAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:13:59.299705Z","signed_message":"canonical_sha256_bytes"},"source_id":"1309.1506","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a248b779677462a44c3bfc879785c9c1f1f202fba384187ba47e9f0d0ae4f9c7","sha256:59d531db8d6d2ba08146a86179ac2d9aec24624bb22fd59268dc65cf49a28e7e"],"state_sha256":"31a85766dc3c0588406d4a14b6ab2bba17453a2c4a0a2f644a3dbc52eabf7482"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HkSf3QZro2getJ7eGaEonVD67KQv3BMK+60pgFneYYl/byvOVIhVhINHoqhcjpxv5PbC1HUV3699RvTCdh/QDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T23:39:17.657074Z","bundle_sha256":"be3508486524dc24be104fd8bac32d7fbdd1c6ed40fbdf60f3fc9d1cc4ecaa09"}}