{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:6DPN2MQVOZTLYWUVUKNBGB5CKO","short_pith_number":"pith:6DPN2MQV","canonical_record":{"source":{"id":"1804.01369","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-04-04T12:40:16Z","cross_cats_sorted":["math.CA","math.GR"],"title_canon_sha256":"e80b6d2a0ecbdbf00983f2197b3be9391b4d01763ba4812c4b4542672a5a1010","abstract_canon_sha256":"074a13bfe8175a7fb885b3344ad863a0d6ed2924b0e7ddacb277fcf9fbb1d2c2"},"schema_version":"1.0"},"canonical_sha256":"f0dedd32157666bc5a95a29a1307a25382375ccfa0605021387a1f487e8b4955","source":{"kind":"arxiv","id":"1804.01369","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.01369","created_at":"2026-05-18T00:04:39Z"},{"alias_kind":"arxiv_version","alias_value":"1804.01369v2","created_at":"2026-05-18T00:04:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.01369","created_at":"2026-05-18T00:04:39Z"},{"alias_kind":"pith_short_12","alias_value":"6DPN2MQVOZTL","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6DPN2MQVOZTLYWUV","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6DPN2MQV","created_at":"2026-05-18T12:32:08Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:6DPN2MQVOZTLYWUVUKNBGB5CKO","target":"record","payload":{"canonical_record":{"source":{"id":"1804.01369","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-04-04T12:40:16Z","cross_cats_sorted":["math.CA","math.GR"],"title_canon_sha256":"e80b6d2a0ecbdbf00983f2197b3be9391b4d01763ba4812c4b4542672a5a1010","abstract_canon_sha256":"074a13bfe8175a7fb885b3344ad863a0d6ed2924b0e7ddacb277fcf9fbb1d2c2"},"schema_version":"1.0"},"canonical_sha256":"f0dedd32157666bc5a95a29a1307a25382375ccfa0605021387a1f487e8b4955","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:39.492940Z","signature_b64":"4i2kGWwJxVTlL/gHV7JfxMJPBMvtWEDdvu16hTSfprLxmVbFG12Vioi/eQRMwYG49DHrx7F3709loxAm94u+CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f0dedd32157666bc5a95a29a1307a25382375ccfa0605021387a1f487e8b4955","last_reissued_at":"2026-05-18T00:04:39.492552Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:39.492552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.01369","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:04:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"P0ObLtmkrJ8l6um0kX28EKxuKJMXeERNiNIjYsUFk6YMTFDVkipFNbEkQsnvKcoRVOaxpvmB+pwbyYi1SBy5AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T11:21:23.365341Z"},"content_sha256":"7bbec5b9160601906dce5b05350289e620359d94993965d370cf586a3b3b69d8","schema_version":"1.0","event_id":"sha256:7bbec5b9160601906dce5b05350289e620359d94993965d370cf586a3b3b69d8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:6DPN2MQVOZTLYWUVUKNBGB5CKO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.GR"],"primary_cat":"math.DS","authors_text":"Catalin Badea, Sophie Grivaux","submitted_at":"2018-04-04T12:40:16Z","abstract_excerpt":"Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\\dots,p_{r}$ there exists a continuous probability measure $\\mu $ on the unit circle $\\mathbb{T}$ such that \\[ \\inf_{k_{1}\\ge 0,\\dots,k_{r}\\ge 0}|\\widehat{\\mu }(p_{1}^{k_{1}}\\dots p_{r}^{k_{r}})|>0. \\] This results applies in particular to the Furstenberg set $F=\\{2^{k}3^{k'}\\,;\\,k\\ge 0,\\ k'\\ge 0\\}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $\\times 2$-$\\times 3$ conjecture. We also estimate the modified Kazhdan const"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01369","kind":"arxiv","version":2},"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":"2026-05-18T00:04:39Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"b0u7hvIw38P7VS44tLgDJlXRNk+jdwwPY6NUO+I3iL1pGLyEsKTuOIjkEL2tuZC/ggKRVaBIk/OBTo+eIIfbAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-24T11:21:23.365719Z"},"content_sha256":"d3c532fb6b4c7e854d4ae2f8f582e5e9e21d6707c73c45009aac225af4c69ff7","schema_version":"1.0","event_id":"sha256:d3c532fb6b4c7e854d4ae2f8f582e5e9e21d6707c73c45009aac225af4c69ff7"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/6DPN2MQVOZTLYWUVUKNBGB5CKO/bundle.json","state_url":"https://pith.science/pith/6DPN2MQVOZTLYWUVUKNBGB5CKO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/6DPN2MQVOZTLYWUVUKNBGB5CKO/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":"2026-06-24T11:21:23Z","links":{"resolver":"https://pith.science/pith/6DPN2MQVOZTLYWUVUKNBGB5CKO","bundle":"https://pith.science/pith/6DPN2MQVOZTLYWUVUKNBGB5CKO/bundle.json","state":"https://pith.science/pith/6DPN2MQVOZTLYWUVUKNBGB5CKO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/6DPN2MQVOZTLYWUVUKNBGB5CKO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:6DPN2MQVOZTLYWUVUKNBGB5CKO","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":"074a13bfe8175a7fb885b3344ad863a0d6ed2924b0e7ddacb277fcf9fbb1d2c2","cross_cats_sorted":["math.CA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-04-04T12:40:16Z","title_canon_sha256":"e80b6d2a0ecbdbf00983f2197b3be9391b4d01763ba4812c4b4542672a5a1010"},"schema_version":"1.0","source":{"id":"1804.01369","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.01369","created_at":"2026-05-18T00:04:39Z"},{"alias_kind":"arxiv_version","alias_value":"1804.01369v2","created_at":"2026-05-18T00:04:39Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.01369","created_at":"2026-05-18T00:04:39Z"},{"alias_kind":"pith_short_12","alias_value":"6DPN2MQVOZTL","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_16","alias_value":"6DPN2MQVOZTLYWUV","created_at":"2026-05-18T12:32:08Z"},{"alias_kind":"pith_short_8","alias_value":"6DPN2MQV","created_at":"2026-05-18T12:32:08Z"}],"graph_snapshots":[{"event_id":"sha256:d3c532fb6b4c7e854d4ae2f8f582e5e9e21d6707c73c45009aac225af4c69ff7","target":"graph","created_at":"2026-05-18T00:04:39Z","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":"Exploiting a construction of rigidity sequences for weakly mixing dynamical systems by Fayad and Thouvenot, we show that for every integers $p_{1},\\dots,p_{r}$ there exists a continuous probability measure $\\mu $ on the unit circle $\\mathbb{T}$ such that \\[ \\inf_{k_{1}\\ge 0,\\dots,k_{r}\\ge 0}|\\widehat{\\mu }(p_{1}^{k_{1}}\\dots p_{r}^{k_{r}})|>0. \\] This results applies in particular to the Furstenberg set $F=\\{2^{k}3^{k'}\\,;\\,k\\ge 0,\\ k'\\ge 0\\}$, and disproves a 1988 conjecture of Lyons inspired by Furstenberg's famous $\\times 2$-$\\times 3$ conjecture. We also estimate the modified Kazhdan const","authors_text":"Catalin Badea, Sophie Grivaux","cross_cats":["math.CA","math.GR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-04-04T12:40:16Z","title":"Kazhdan constants, continuous probability measures with large Fourier coefficients and rigidity sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.01369","kind":"arxiv","version":2},"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":"sha256:7bbec5b9160601906dce5b05350289e620359d94993965d370cf586a3b3b69d8","target":"record","created_at":"2026-05-18T00:04:39Z","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":"074a13bfe8175a7fb885b3344ad863a0d6ed2924b0e7ddacb277fcf9fbb1d2c2","cross_cats_sorted":["math.CA","math.GR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2018-04-04T12:40:16Z","title_canon_sha256":"e80b6d2a0ecbdbf00983f2197b3be9391b4d01763ba4812c4b4542672a5a1010"},"schema_version":"1.0","source":{"id":"1804.01369","kind":"arxiv","version":2}},"canonical_sha256":"f0dedd32157666bc5a95a29a1307a25382375ccfa0605021387a1f487e8b4955","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f0dedd32157666bc5a95a29a1307a25382375ccfa0605021387a1f487e8b4955","first_computed_at":"2026-05-18T00:04:39.492552Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:04:39.492552Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"4i2kGWwJxVTlL/gHV7JfxMJPBMvtWEDdvu16hTSfprLxmVbFG12Vioi/eQRMwYG49DHrx7F3709loxAm94u+CQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:04:39.492940Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.01369","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7bbec5b9160601906dce5b05350289e620359d94993965d370cf586a3b3b69d8","sha256:d3c532fb6b4c7e854d4ae2f8f582e5e9e21d6707c73c45009aac225af4c69ff7"],"state_sha256":"12ea73d5a990f396e95275e42eee1b72ee029754150f4722afc80f9eac2e7851"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kkcH0jCjiVc5jMJfdiPWbW/5JUAxpvPHyflGXcpys0o1L6B8T+IAkzP6UfcS79X84BaveZGApfzTXuxf/UMvAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-24T11:21:23.367986Z","bundle_sha256":"ca655f5ab3dd122cd2f4263acac0952dae53ed3170d48b57036f48f63c5d8747"}}