{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:EHYVOAFIH2SMKSEI4H3YTI5GGV","short_pith_number":"pith:EHYVOAFI","canonical_record":{"source":{"id":"1311.4926","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-20T00:33:58Z","cross_cats_sorted":[],"title_canon_sha256":"863f5e853f7cf5630bd584018aef0dac30de26facf0e2d3e91ef4aed3e4b3b33","abstract_canon_sha256":"4ab2949706bcfec66b05650e22941cdc19cabb93f8a3f553d224079d173e7b72"},"schema_version":"1.0"},"canonical_sha256":"21f15700a83ea4c54888e1f789a3a63573197bf7d4fd0c514f93c3da711aa2e2","source":{"kind":"arxiv","id":"1311.4926","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4926","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4926v2","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4926","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"EHYVOAFIH2SM","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EHYVOAFIH2SMKSEI","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EHYVOAFI","created_at":"2026-05-18T12:27:43Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:EHYVOAFIH2SMKSEI4H3YTI5GGV","target":"record","payload":{"canonical_record":{"source":{"id":"1311.4926","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-20T00:33:58Z","cross_cats_sorted":[],"title_canon_sha256":"863f5e853f7cf5630bd584018aef0dac30de26facf0e2d3e91ef4aed3e4b3b33","abstract_canon_sha256":"4ab2949706bcfec66b05650e22941cdc19cabb93f8a3f553d224079d173e7b72"},"schema_version":"1.0"},"canonical_sha256":"21f15700a83ea4c54888e1f789a3a63573197bf7d4fd0c514f93c3da711aa2e2","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:02:46.511348Z","signature_b64":"yZIMM+BZDChg4Oe/SjAOaOjWu9JJHTr84ky/VC2cFzqZ/k5j6kUz55rBe4ExczZ5nzQ5LqDwTqIOuTZFtY5ECg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"21f15700a83ea4c54888e1f789a3a63573197bf7d4fd0c514f93c3da711aa2e2","last_reissued_at":"2026-05-18T03:02:46.510716Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:02:46.510716Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1311.4926","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":"2026-05-18T03:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"uVHq6x7OSQ8KgkEWrY/OKts8yRcohhQ/MpM/XRGytOL8I5xmixSEotsYXkc3PgG+bSEhYWVT+X9e7T0pi7fyAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:47:01.799629Z"},"content_sha256":"7490df696821184fd62bea82f7e52cf4a65186a02146b69e9570e2a66f53f730","schema_version":"1.0","event_id":"sha256:7490df696821184fd62bea82f7e52cf4a65186a02146b69e9570e2a66f53f730"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:EHYVOAFIH2SMKSEI4H3YTI5GGV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the system $f(nx)$ and probabilistic number theory","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christoph Aistleitner, Istvan Berkes, Robert Tichy","submitted_at":"2013-11-20T00:33:58Z","abstract_excerpt":"Let $f: {\\mathbb R}\\to {\\mathbb R}$ be a measurable function satisfying \\begin{equation*} f(x+1)=f(x), \\qquad \\int_0^1 f(x)\\, dx=0, \\qquad \\int_0^1 f^2(x)\\, dx<\\infty. \\end{equation*} The asymptotic properties of series $\\sum c_k f(kx)$ have been studied extensively in the literature and turned out to be, in general, quite different from those of the trigonometric system. As the theory shows, the behavior of such series is determined by a combination of analytic, probabilistic and number theoretic effects, resulting in highly interesting phenomena not encountered in classical harmonic analysis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4926","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":"2026-05-18T03:02:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H5TUGMQO1Gw4R0xIpnfSin01OLlE7sCclAq51KiK4md69gev78YlePFsdKdDoTSL4fFB8fJXZmSm6/3rMMreCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T10:47:01.799968Z"},"content_sha256":"176cdebd8b21ecc65f71e91241057b268eb35871b3738f37c45fa6169a2f2184","schema_version":"1.0","event_id":"sha256:176cdebd8b21ecc65f71e91241057b268eb35871b3738f37c45fa6169a2f2184"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EHYVOAFIH2SMKSEI4H3YTI5GGV/bundle.json","state_url":"https://pith.science/pith/EHYVOAFIH2SMKSEI4H3YTI5GGV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EHYVOAFIH2SMKSEI4H3YTI5GGV/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":"2026-06-01T10:47:01Z","links":{"resolver":"https://pith.science/pith/EHYVOAFIH2SMKSEI4H3YTI5GGV","bundle":"https://pith.science/pith/EHYVOAFIH2SMKSEI4H3YTI5GGV/bundle.json","state":"https://pith.science/pith/EHYVOAFIH2SMKSEI4H3YTI5GGV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EHYVOAFIH2SMKSEI4H3YTI5GGV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:EHYVOAFIH2SMKSEI4H3YTI5GGV","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":"4ab2949706bcfec66b05650e22941cdc19cabb93f8a3f553d224079d173e7b72","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-20T00:33:58Z","title_canon_sha256":"863f5e853f7cf5630bd584018aef0dac30de26facf0e2d3e91ef4aed3e4b3b33"},"schema_version":"1.0","source":{"id":"1311.4926","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1311.4926","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"arxiv_version","alias_value":"1311.4926v2","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1311.4926","created_at":"2026-05-18T03:02:46Z"},{"alias_kind":"pith_short_12","alias_value":"EHYVOAFIH2SM","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_16","alias_value":"EHYVOAFIH2SMKSEI","created_at":"2026-05-18T12:27:43Z"},{"alias_kind":"pith_short_8","alias_value":"EHYVOAFI","created_at":"2026-05-18T12:27:43Z"}],"graph_snapshots":[{"event_id":"sha256:176cdebd8b21ecc65f71e91241057b268eb35871b3738f37c45fa6169a2f2184","target":"graph","created_at":"2026-05-18T03:02:46Z","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 $f: {\\mathbb R}\\to {\\mathbb R}$ be a measurable function satisfying \\begin{equation*} f(x+1)=f(x), \\qquad \\int_0^1 f(x)\\, dx=0, \\qquad \\int_0^1 f^2(x)\\, dx<\\infty. \\end{equation*} The asymptotic properties of series $\\sum c_k f(kx)$ have been studied extensively in the literature and turned out to be, in general, quite different from those of the trigonometric system. As the theory shows, the behavior of such series is determined by a combination of analytic, probabilistic and number theoretic effects, resulting in highly interesting phenomena not encountered in classical harmonic analysis","authors_text":"Christoph Aistleitner, Istvan Berkes, Robert Tichy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-20T00:33:58Z","title":"On the system $f(nx)$ and probabilistic number theory"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.4926","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":"sha256:7490df696821184fd62bea82f7e52cf4a65186a02146b69e9570e2a66f53f730","target":"record","created_at":"2026-05-18T03:02:46Z","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":"4ab2949706bcfec66b05650e22941cdc19cabb93f8a3f553d224079d173e7b72","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-11-20T00:33:58Z","title_canon_sha256":"863f5e853f7cf5630bd584018aef0dac30de26facf0e2d3e91ef4aed3e4b3b33"},"schema_version":"1.0","source":{"id":"1311.4926","kind":"arxiv","version":2}},"canonical_sha256":"21f15700a83ea4c54888e1f789a3a63573197bf7d4fd0c514f93c3da711aa2e2","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"21f15700a83ea4c54888e1f789a3a63573197bf7d4fd0c514f93c3da711aa2e2","first_computed_at":"2026-05-18T03:02:46.510716Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:02:46.510716Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yZIMM+BZDChg4Oe/SjAOaOjWu9JJHTr84ky/VC2cFzqZ/k5j6kUz55rBe4ExczZ5nzQ5LqDwTqIOuTZFtY5ECg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:02:46.511348Z","signed_message":"canonical_sha256_bytes"},"source_id":"1311.4926","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7490df696821184fd62bea82f7e52cf4a65186a02146b69e9570e2a66f53f730","sha256:176cdebd8b21ecc65f71e91241057b268eb35871b3738f37c45fa6169a2f2184"],"state_sha256":"b699ef8a88e909f20e56d4a823d3cf928de38182d1543ff86032b4f8821e4c6c"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"GztTH45f33l/YOmkstUIA1fEQx70nIEPckLcuoHxBhlSHVKwGuueJygWKeg0wtHijnfnUOGI1mgiPnOasnrABw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T10:47:01.801849Z","bundle_sha256":"a78bc3401ef4a9723cd17232c0cd874c0a9e35683000356fe4fc38d992e0fd0b"}}