{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:56FXK6KDWU3MANGYYTUW4WHJLR","short_pith_number":"pith:56FXK6KD","canonical_record":{"source":{"id":"1703.03780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-08T19:43:10Z","cross_cats_sorted":[],"title_canon_sha256":"405d56965a952a173e26a43e017c8f082c61b50dc310864c24bb464e1866b1fb","abstract_canon_sha256":"f9c081c703d1646505e3abe7dff2c84137bd5e260d54148c93e176e462c3feef"},"schema_version":"1.0"},"canonical_sha256":"ef8b757943b536c034d8c4e96e58e95c5a0d972a2a3bf7c909a93fb0e7589641","source":{"kind":"arxiv","id":"1703.03780","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.03780","created_at":"2026-05-18T00:48:55Z"},{"alias_kind":"arxiv_version","alias_value":"1703.03780v1","created_at":"2026-05-18T00:48:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.03780","created_at":"2026-05-18T00:48:55Z"},{"alias_kind":"pith_short_12","alias_value":"56FXK6KDWU3M","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"56FXK6KDWU3MANGY","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"56FXK6KD","created_at":"2026-05-18T12:31:00Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:56FXK6KDWU3MANGYYTUW4WHJLR","target":"record","payload":{"canonical_record":{"source":{"id":"1703.03780","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-08T19:43:10Z","cross_cats_sorted":[],"title_canon_sha256":"405d56965a952a173e26a43e017c8f082c61b50dc310864c24bb464e1866b1fb","abstract_canon_sha256":"f9c081c703d1646505e3abe7dff2c84137bd5e260d54148c93e176e462c3feef"},"schema_version":"1.0"},"canonical_sha256":"ef8b757943b536c034d8c4e96e58e95c5a0d972a2a3bf7c909a93fb0e7589641","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:48:55.768348Z","signature_b64":"wuAbWVN9TNSzDV4yf9GqdR9akuj/gP+qmA7O9Zk8/BNNmN23Z2vODZOLYroTHB9nBlmAPUyLGhAcARvtxot4DA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ef8b757943b536c034d8c4e96e58e95c5a0d972a2a3bf7c909a93fb0e7589641","last_reissued_at":"2026-05-18T00:48:55.767585Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:48:55.767585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1703.03780","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":"2026-05-18T00:48:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Pfm7+UNzMkyAcgyQN/buo0dywX8wwrgsgN4Vm4ek8IyO3VrlMf4eOZy/ZXafuCjetMqerrNeJYHjFN8iI8O0BA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T13:50:44.825859Z"},"content_sha256":"40bbde9a38a05d18af1437b95dc865e5aa415722afddf9d9b86556390ec65682","schema_version":"1.0","event_id":"sha256:40bbde9a38a05d18af1437b95dc865e5aa415722afddf9d9b86556390ec65682"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:56FXK6KDWU3MANGYYTUW4WHJLR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Lacunary arithmetic statistical convergence","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GM","authors_text":"Bipan Hazarika, Taja Yaying","submitted_at":"2017-03-08T19:43:10Z","abstract_excerpt":"A lacunary sequence is an increasing integer sequence $\\theta=(k_r)$ such that $k_r-k_{r-1}\\rightarrow \\infty$ as $r\\rightarrow \\infty.$ In this article we introduce arithmetic statistically convergent sequence space $ASC$ and lacunary arithmetic statistically convergent sequence space $ASC_{\\theta}$ and study some inclusion properties between the two spaces. Finally we introduce lacunary arithmetic statistical continuity and establish some interesting results."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.03780","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":"2026-05-18T00:48:55Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1f2DKkeHbUi607RIzb8PjJRKDI7qLZH4fVzxHc//bAvQYJW3024XFm25EofLSFpuIad2dsAJ3j4QkxNAEHDMBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-02T13:50:44.826193Z"},"content_sha256":"b7d033791b86b1d46638a88359bd46e6d20b26b162a02f517a2e662f567d7d6f","schema_version":"1.0","event_id":"sha256:b7d033791b86b1d46638a88359bd46e6d20b26b162a02f517a2e662f567d7d6f"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/56FXK6KDWU3MANGYYTUW4WHJLR/bundle.json","state_url":"https://pith.science/pith/56FXK6KDWU3MANGYYTUW4WHJLR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/56FXK6KDWU3MANGYYTUW4WHJLR/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":"2026-07-02T13:50:44Z","links":{"resolver":"https://pith.science/pith/56FXK6KDWU3MANGYYTUW4WHJLR","bundle":"https://pith.science/pith/56FXK6KDWU3MANGYYTUW4WHJLR/bundle.json","state":"https://pith.science/pith/56FXK6KDWU3MANGYYTUW4WHJLR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/56FXK6KDWU3MANGYYTUW4WHJLR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:56FXK6KDWU3MANGYYTUW4WHJLR","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":"f9c081c703d1646505e3abe7dff2c84137bd5e260d54148c93e176e462c3feef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-08T19:43:10Z","title_canon_sha256":"405d56965a952a173e26a43e017c8f082c61b50dc310864c24bb464e1866b1fb"},"schema_version":"1.0","source":{"id":"1703.03780","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1703.03780","created_at":"2026-05-18T00:48:55Z"},{"alias_kind":"arxiv_version","alias_value":"1703.03780v1","created_at":"2026-05-18T00:48:55Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1703.03780","created_at":"2026-05-18T00:48:55Z"},{"alias_kind":"pith_short_12","alias_value":"56FXK6KDWU3M","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_16","alias_value":"56FXK6KDWU3MANGY","created_at":"2026-05-18T12:31:00Z"},{"alias_kind":"pith_short_8","alias_value":"56FXK6KD","created_at":"2026-05-18T12:31:00Z"}],"graph_snapshots":[{"event_id":"sha256:b7d033791b86b1d46638a88359bd46e6d20b26b162a02f517a2e662f567d7d6f","target":"graph","created_at":"2026-05-18T00:48:55Z","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":"A lacunary sequence is an increasing integer sequence $\\theta=(k_r)$ such that $k_r-k_{r-1}\\rightarrow \\infty$ as $r\\rightarrow \\infty.$ In this article we introduce arithmetic statistically convergent sequence space $ASC$ and lacunary arithmetic statistically convergent sequence space $ASC_{\\theta}$ and study some inclusion properties between the two spaces. Finally we introduce lacunary arithmetic statistical continuity and establish some interesting results.","authors_text":"Bipan Hazarika, Taja Yaying","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-08T19:43:10Z","title":"Lacunary arithmetic statistical convergence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1703.03780","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":"sha256:40bbde9a38a05d18af1437b95dc865e5aa415722afddf9d9b86556390ec65682","target":"record","created_at":"2026-05-18T00:48:55Z","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":"f9c081c703d1646505e3abe7dff2c84137bd5e260d54148c93e176e462c3feef","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GM","submitted_at":"2017-03-08T19:43:10Z","title_canon_sha256":"405d56965a952a173e26a43e017c8f082c61b50dc310864c24bb464e1866b1fb"},"schema_version":"1.0","source":{"id":"1703.03780","kind":"arxiv","version":1}},"canonical_sha256":"ef8b757943b536c034d8c4e96e58e95c5a0d972a2a3bf7c909a93fb0e7589641","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ef8b757943b536c034d8c4e96e58e95c5a0d972a2a3bf7c909a93fb0e7589641","first_computed_at":"2026-05-18T00:48:55.767585Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:48:55.767585Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wuAbWVN9TNSzDV4yf9GqdR9akuj/gP+qmA7O9Zk8/BNNmN23Z2vODZOLYroTHB9nBlmAPUyLGhAcARvtxot4DA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:48:55.768348Z","signed_message":"canonical_sha256_bytes"},"source_id":"1703.03780","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:40bbde9a38a05d18af1437b95dc865e5aa415722afddf9d9b86556390ec65682","sha256:b7d033791b86b1d46638a88359bd46e6d20b26b162a02f517a2e662f567d7d6f"],"state_sha256":"0d8b1c199a2e81a29b9df11862d60861c426e29eadd982ee77487c2ab0d26272"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/ZD1cJgJHRYz46I3fzNj2Qhyfl76ufdzHeaWxgQM0PrQ4imAd254V2/fiGbXrJJzaa4fjmJ9W0ylMtrwZKXoAQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-02T13:50:44.828024Z","bundle_sha256":"110f33dee8efe363270644618db83a702de17a6e9a84d80c8f77af9a0268f7d8"}}