{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:UNYEICDWOMYMEDOTFRIT74BLEV","short_pith_number":"pith:UNYEICDW","canonical_record":{"source":{"id":"1804.07560","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-20T11:44:14Z","cross_cats_sorted":[],"title_canon_sha256":"f99e4eddf2b878f018cec90afd228a21856c3867da0db0c9e79cd83a0a328cd6","abstract_canon_sha256":"38f0ff447376a7551f21e78507121637befbd0dffea3316399b2b999d824bc0c"},"schema_version":"1.0"},"canonical_sha256":"a3704408767330c20dd32c513ff02b256ad5cd15ba52d205d48e8df10fea8237","source":{"kind":"arxiv","id":"1804.07560","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.07560","created_at":"2026-05-18T00:17:58Z"},{"alias_kind":"arxiv_version","alias_value":"1804.07560v1","created_at":"2026-05-18T00:17:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.07560","created_at":"2026-05-18T00:17:58Z"},{"alias_kind":"pith_short_12","alias_value":"UNYEICDWOMYM","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"UNYEICDWOMYMEDOT","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"UNYEICDW","created_at":"2026-05-18T12:32:56Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:UNYEICDWOMYMEDOTFRIT74BLEV","target":"record","payload":{"canonical_record":{"source":{"id":"1804.07560","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-20T11:44:14Z","cross_cats_sorted":[],"title_canon_sha256":"f99e4eddf2b878f018cec90afd228a21856c3867da0db0c9e79cd83a0a328cd6","abstract_canon_sha256":"38f0ff447376a7551f21e78507121637befbd0dffea3316399b2b999d824bc0c"},"schema_version":"1.0"},"canonical_sha256":"a3704408767330c20dd32c513ff02b256ad5cd15ba52d205d48e8df10fea8237","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:58.459810Z","signature_b64":"xfLShzdUGP6tRNuJw6Tfe9v1gDV/hFOnUjn2ogMv1GiSqY9VvquJ0rIjO/cqHnFCabqQ4CyT/VZ38A81g6UBDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"a3704408767330c20dd32c513ff02b256ad5cd15ba52d205d48e8df10fea8237","last_reissued_at":"2026-05-18T00:17:58.459357Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:58.459357Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1804.07560","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":"2026-05-18T00:17:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"U8o7+q9mwOl8pUhJMXQqPXTkr0WJNVdjGl4TWzlGkTdSr0Pb008SY3hDrZcVX3jVBNvU1dZ88HeuGD2b70eqCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:57:02.654086Z"},"content_sha256":"c80cb77689257eb95c07a43fc984523dd8d46d4751f2e5cfc58384fcedbf7ced","schema_version":"1.0","event_id":"sha256:c80cb77689257eb95c07a43fc984523dd8d46d4751f2e5cfc58384fcedbf7ced"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:UNYEICDWOMYMEDOTFRIT74BLEV","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Generalizations of some results about the regularity properties of an additive representation function","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Csaba S\\'andor, S\\'andor Z. Kiss","submitted_at":"2018-04-20T11:44:14Z","abstract_excerpt":"Let $A = \\{a_{1},a_{2},\\dots{}\\}$ $(a_{1} < a_{2} < \\dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\\in A)$. P. Erd\\H{o}s, A. S\\'ark\\\"ozy and V. T. S\\'os proved that if $\\lim_{N\\to\\infty}\\frac{B(A,N)}{\\sqrt{N}}=+\\infty$ then $|\\Delta_{1}(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and $\\Delta_{l}$ denotes the $l$-th difference. Their result was extended to $\\Delta_{l}(R_{A,2}(n))$ for any fixed $l\\ge2$. In this paper we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07560","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":"2026-05-18T00:17:58Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZMlaFjFhGkfDziIpUar1YaTxFPhpJox41u253UuU+EXJQ/vTnxcHYQiqWRbQeGhyuOqaqu/b0hxXPbdabbI3DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T14:57:02.654452Z"},"content_sha256":"e6c497e8d2dba886478e583ba0ef38eed2d987cdf244de060ab2a8f0fa38890a","schema_version":"1.0","event_id":"sha256:e6c497e8d2dba886478e583ba0ef38eed2d987cdf244de060ab2a8f0fa38890a"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/UNYEICDWOMYMEDOTFRIT74BLEV/bundle.json","state_url":"https://pith.science/pith/UNYEICDWOMYMEDOTFRIT74BLEV/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/UNYEICDWOMYMEDOTFRIT74BLEV/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":"2026-06-22T14:57:02Z","links":{"resolver":"https://pith.science/pith/UNYEICDWOMYMEDOTFRIT74BLEV","bundle":"https://pith.science/pith/UNYEICDWOMYMEDOTFRIT74BLEV/bundle.json","state":"https://pith.science/pith/UNYEICDWOMYMEDOTFRIT74BLEV/state.json","well_known_bundle":"https://pith.science/.well-known/pith/UNYEICDWOMYMEDOTFRIT74BLEV/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:UNYEICDWOMYMEDOTFRIT74BLEV","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":"38f0ff447376a7551f21e78507121637befbd0dffea3316399b2b999d824bc0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-20T11:44:14Z","title_canon_sha256":"f99e4eddf2b878f018cec90afd228a21856c3867da0db0c9e79cd83a0a328cd6"},"schema_version":"1.0","source":{"id":"1804.07560","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1804.07560","created_at":"2026-05-18T00:17:58Z"},{"alias_kind":"arxiv_version","alias_value":"1804.07560v1","created_at":"2026-05-18T00:17:58Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1804.07560","created_at":"2026-05-18T00:17:58Z"},{"alias_kind":"pith_short_12","alias_value":"UNYEICDWOMYM","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_16","alias_value":"UNYEICDWOMYMEDOT","created_at":"2026-05-18T12:32:56Z"},{"alias_kind":"pith_short_8","alias_value":"UNYEICDW","created_at":"2026-05-18T12:32:56Z"}],"graph_snapshots":[{"event_id":"sha256:e6c497e8d2dba886478e583ba0ef38eed2d987cdf244de060ab2a8f0fa38890a","target":"graph","created_at":"2026-05-18T00:17:58Z","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 $A = \\{a_{1},a_{2},\\dots{}\\}$ $(a_{1} < a_{2} < \\dots{})$ be an infinite sequence of nonnegative integers, and let $R_{A,2}(n)$ denote the number of solutions of $a_{x}+a_{y}=n$ $(a_{x},a_{y}\\in A)$. P. Erd\\H{o}s, A. S\\'ark\\\"ozy and V. T. S\\'os proved that if $\\lim_{N\\to\\infty}\\frac{B(A,N)}{\\sqrt{N}}=+\\infty$ then $|\\Delta_{1}(R_{A,2}(n))|$ cannot be bounded, where $B(A,N)$ denotes the number of blocks formed by consecutive integers in $A$ up to $N$ and $\\Delta_{l}$ denotes the $l$-th difference. Their result was extended to $\\Delta_{l}(R_{A,2}(n))$ for any fixed $l\\ge2$. In this paper we ","authors_text":"Csaba S\\'andor, S\\'andor Z. Kiss","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-20T11:44:14Z","title":"Generalizations of some results about the regularity properties of an additive representation function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.07560","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:c80cb77689257eb95c07a43fc984523dd8d46d4751f2e5cfc58384fcedbf7ced","target":"record","created_at":"2026-05-18T00:17:58Z","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":"38f0ff447376a7551f21e78507121637befbd0dffea3316399b2b999d824bc0c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-20T11:44:14Z","title_canon_sha256":"f99e4eddf2b878f018cec90afd228a21856c3867da0db0c9e79cd83a0a328cd6"},"schema_version":"1.0","source":{"id":"1804.07560","kind":"arxiv","version":1}},"canonical_sha256":"a3704408767330c20dd32c513ff02b256ad5cd15ba52d205d48e8df10fea8237","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a3704408767330c20dd32c513ff02b256ad5cd15ba52d205d48e8df10fea8237","first_computed_at":"2026-05-18T00:17:58.459357Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:17:58.459357Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"xfLShzdUGP6tRNuJw6Tfe9v1gDV/hFOnUjn2ogMv1GiSqY9VvquJ0rIjO/cqHnFCabqQ4CyT/VZ38A81g6UBDA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:17:58.459810Z","signed_message":"canonical_sha256_bytes"},"source_id":"1804.07560","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c80cb77689257eb95c07a43fc984523dd8d46d4751f2e5cfc58384fcedbf7ced","sha256:e6c497e8d2dba886478e583ba0ef38eed2d987cdf244de060ab2a8f0fa38890a"],"state_sha256":"4dba0edd132c0d3b7be7da11b289e0347a843dbe704e092f3d3e2ba253c19ad8"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"fi9kiYPbwNApxDeil1RojrhDy+1PtAmEQrAJSsl2bcCNPoRKVShtTQ1kVc+yfGhTAbLmS1mLoi8Qrn1xrRYhAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T14:57:02.656408Z","bundle_sha256":"f534e81f4e435b4e0ce6b09350bf7a8ee349615b2cdc19557168dd8341ea6433"}}