{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2025:3J45EFMHL2NZ6Y3Y23API6EH4V","short_pith_number":"pith:3J45EFMH","canonical_record":{"source":{"id":"2504.00708","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-04-01T12:17:23Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"c9ec2e17e234daf7df2d5e01448f864c808820121fa756c36fe037558f941ec9","abstract_canon_sha256":"868c0882d4adb234b165b98df55c111885df42efc957b49e25eeae75da03ac52"},"schema_version":"1.0"},"canonical_sha256":"da79d215875e9b9f6378d6c0f47887e55dddf3b82ace687f6934919049b9c7f2","source":{"kind":"arxiv","id":"2504.00708","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.00708","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"arxiv_version","alias_value":"2504.00708v1","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.00708","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"pith_short_12","alias_value":"3J45EFMHL2NZ","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"pith_short_16","alias_value":"3J45EFMHL2NZ6Y3Y","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"pith_short_8","alias_value":"3J45EFMH","created_at":"2026-07-05T10:42:49Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2025:3J45EFMHL2NZ6Y3Y23API6EH4V","target":"record","payload":{"canonical_record":{"source":{"id":"2504.00708","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-04-01T12:17:23Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"c9ec2e17e234daf7df2d5e01448f864c808820121fa756c36fe037558f941ec9","abstract_canon_sha256":"868c0882d4adb234b165b98df55c111885df42efc957b49e25eeae75da03ac52"},"schema_version":"1.0"},"canonical_sha256":"da79d215875e9b9f6378d6c0f47887e55dddf3b82ace687f6934919049b9c7f2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:42:49.160866Z","signature_b64":"LPnD6ltMqNSdzUNs8HwDKWe+1Tpu0bZPvthwrHqzt3Af+s0CadQ8fPC9BIwTzOWGok3spzhCM7Fbu+x8AsFmBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"da79d215875e9b9f6378d6c0f47887e55dddf3b82ace687f6934919049b9c7f2","last_reissued_at":"2026-07-05T10:42:49.160361Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:42:49.160361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2504.00708","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":"2026-07-05T10:42:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"I5/9PeucsDkOCZEfqyFt3SMor84zhEhx+GOlyQs5FkVdY3Y5xarHKcxAzB1sLK5yXySR5Ag6YbHuQVgJwSkMDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T10:15:52.306505Z"},"content_sha256":"bd2412f50d0cdee85f9de775247c6b91ecd5188cccb0a80ed095932f0e545e8b","schema_version":"1.0","event_id":"sha256:bd2412f50d0cdee85f9de775247c6b91ecd5188cccb0a80ed095932f0e545e8b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2025:3J45EFMHL2NZ6Y3Y23API6EH4V","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Almost sure asymptotics for the number variance of dilations of integer sequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.NT","authors_text":"Christoph Aistleitner, Nadav Yesha","submitted_at":"2025-04-01T12:17:23Z","abstract_excerpt":"Let $(x_n)_{n=1}^\\infty$ be a sequence of integers. We study the number variance of dilations $(\\alpha x_n)_{n=1}^\\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $\\alpha$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $\\alpha$ throughout the range $0 \\leq S \\leq (\\log N)^{-c}$, for a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.00708","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2504.00708/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T10:42:49Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"or+uAUuvVrk9sJRjXtuPvEAJie8xLWrKcix8Ofpi1ZSkkDTy8udNJvhzkGKVGmlL6gaPj57iLF5wFprkpdx9DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-08T10:15:52.307065Z"},"content_sha256":"b07cbd174abf70702f716736d1235e3ca7c1bf33ce025ec3571b64524faaa6fb","schema_version":"1.0","event_id":"sha256:b07cbd174abf70702f716736d1235e3ca7c1bf33ce025ec3571b64524faaa6fb"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/3J45EFMHL2NZ6Y3Y23API6EH4V/bundle.json","state_url":"https://pith.science/pith/3J45EFMHL2NZ6Y3Y23API6EH4V/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/3J45EFMHL2NZ6Y3Y23API6EH4V/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":"2026-08-08T10:15:52Z","links":{"resolver":"https://pith.science/pith/3J45EFMHL2NZ6Y3Y23API6EH4V","bundle":"https://pith.science/pith/3J45EFMHL2NZ6Y3Y23API6EH4V/bundle.json","state":"https://pith.science/pith/3J45EFMHL2NZ6Y3Y23API6EH4V/state.json","well_known_bundle":"https://pith.science/.well-known/pith/3J45EFMHL2NZ6Y3Y23API6EH4V/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2025:3J45EFMHL2NZ6Y3Y23API6EH4V","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":"868c0882d4adb234b165b98df55c111885df42efc957b49e25eeae75da03ac52","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-04-01T12:17:23Z","title_canon_sha256":"c9ec2e17e234daf7df2d5e01448f864c808820121fa756c36fe037558f941ec9"},"schema_version":"1.0","source":{"id":"2504.00708","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2504.00708","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"arxiv_version","alias_value":"2504.00708v1","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.00708","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"pith_short_12","alias_value":"3J45EFMHL2NZ","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"pith_short_16","alias_value":"3J45EFMHL2NZ6Y3Y","created_at":"2026-07-05T10:42:49Z"},{"alias_kind":"pith_short_8","alias_value":"3J45EFMH","created_at":"2026-07-05T10:42:49Z"}],"graph_snapshots":[{"event_id":"sha256:b07cbd174abf70702f716736d1235e3ca7c1bf33ce025ec3571b64524faaa6fb","target":"graph","created_at":"2026-07-05T10:42:49Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2504.00708/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $(x_n)_{n=1}^\\infty$ be a sequence of integers. We study the number variance of dilations $(\\alpha x_n)_{n=1}^\\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $\\alpha$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $\\alpha$ throughout the range $0 \\leq S \\leq (\\log N)^{-c}$, for a ","authors_text":"Christoph Aistleitner, Nadav Yesha","cross_cats":["math.PR"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-04-01T12:17:23Z","title":"Almost sure asymptotics for the number variance of dilations of integer sequences"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.00708","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:bd2412f50d0cdee85f9de775247c6b91ecd5188cccb0a80ed095932f0e545e8b","target":"record","created_at":"2026-07-05T10:42:49Z","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":"868c0882d4adb234b165b98df55c111885df42efc957b49e25eeae75da03ac52","cross_cats_sorted":["math.PR"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-04-01T12:17:23Z","title_canon_sha256":"c9ec2e17e234daf7df2d5e01448f864c808820121fa756c36fe037558f941ec9"},"schema_version":"1.0","source":{"id":"2504.00708","kind":"arxiv","version":1}},"canonical_sha256":"da79d215875e9b9f6378d6c0f47887e55dddf3b82ace687f6934919049b9c7f2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"da79d215875e9b9f6378d6c0f47887e55dddf3b82ace687f6934919049b9c7f2","first_computed_at":"2026-07-05T10:42:49.160361Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:42:49.160361Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LPnD6ltMqNSdzUNs8HwDKWe+1Tpu0bZPvthwrHqzt3Af+s0CadQ8fPC9BIwTzOWGok3spzhCM7Fbu+x8AsFmBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:42:49.160866Z","signed_message":"canonical_sha256_bytes"},"source_id":"2504.00708","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:bd2412f50d0cdee85f9de775247c6b91ecd5188cccb0a80ed095932f0e545e8b","sha256:b07cbd174abf70702f716736d1235e3ca7c1bf33ce025ec3571b64524faaa6fb"],"state_sha256":"5c3c94e6ce5a280d586c53440518d276c44c79ddf79f1037685589c38a6892de"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9W64YRnjKZeRN+HjiwJ5p25jR/7vxK/OZdVz9ovUeObZPDhUlFR+jfG7Hy2mxBJUx3aXftvT2+B0SC65shoADQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-08T10:15:52.311927Z","bundle_sha256":"e777a28ebf82efaadd9dc6c5b61bcd69df32e9f0b56628cb02ab66ea7577e007"}}