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This means that for any non-negative measurable $f: {{\\mathbb R}}\\to [0,+ {\\infty})$ either the convergence set $C(f, {\\Lambda})=\\{x: s(x)<+ {\\infty} \\}= {{\\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, {\\Lambda})=\\{x: s(x)=+ {\\infty} \\}= {{\\mathbb R}}$ modulo sets of measure zero.\n  If $ {\\Lambda}$ is not type $1$ we say that $ {\\Lambda}$ is type 2.\n  The exact c"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1805.12419","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-05-31T11:17:34Z","cross_cats_sorted":[],"title_canon_sha256":"f3b99de4df8de2932b7f25e57284eb0deabeecd465634c36fd7d2d3bbf4ceadb","abstract_canon_sha256":"9b1f7def1eb3a442f107cc25652db9c47fa63d548751618e18e5d35bbde2adb7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:14:30.246955Z","signature_b64":"SpLpfckOs2RKVZSUI5y3X00WuwpzCGN1Xj25jsKSVj5LcAUh6Kl39tUh/rmzBGbTvqkWMJ90I/AiDfkNrTqjAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9aef2bfee7e95ec5568ee2d16e2573dcfe3147e6082536047463d12c163296a3","last_reissued_at":"2026-05-18T00:14:30.246368Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:14:30.246368Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Type $1$ and $2$ sets for series of translates of functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Bal\\'azs Maga, Bruce Hanson, G\\'asp\\'ar V\\'ertesy, Zolt\\'an Buczolich","submitted_at":"2018-05-31T11:17:34Z","abstract_excerpt":"Suppose $\\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\\Lambda}$ is type $1$ if the series $s(x)=\\sum_{\\lambda\\in\\Lambda}f(x+\\lambda)$ satisfies a zero-one law. This means that for any non-negative measurable $f: {{\\mathbb R}}\\to [0,+ {\\infty})$ either the convergence set $C(f, {\\Lambda})=\\{x: s(x)<+ {\\infty} \\}= {{\\mathbb R}}$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, {\\Lambda})=\\{x: s(x)=+ {\\infty} \\}= {{\\mathbb R}}$ modulo sets of measure zero.\n  If $ {\\Lambda}$ is not type $1$ we say that $ {\\Lambda}$ is type 2.\n  The exact c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.12419","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1805.12419","created_at":"2026-05-18T00:14:30.246458+00:00"},{"alias_kind":"arxiv_version","alias_value":"1805.12419v1","created_at":"2026-05-18T00:14:30.246458+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.12419","created_at":"2026-05-18T00:14:30.246458+00:00"},{"alias_kind":"pith_short_12","alias_value":"TLXSX7XH5FPM","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"TLXSX7XH5FPMKVUO","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"TLXSX7XH","created_at":"2026-05-18T12:32:53.628368+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T","json":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T.json","graph_json":"https://pith.science/api/pith-number/TLXSX7XH5FPMKVUO4LIW4JLT3T/graph.json","events_json":"https://pith.science/api/pith-number/TLXSX7XH5FPMKVUO4LIW4JLT3T/events.json","paper":"https://pith.science/paper/TLXSX7XH"},"agent_actions":{"view_html":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T","download_json":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T.json","view_paper":"https://pith.science/paper/TLXSX7XH","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1805.12419&json=true","fetch_graph":"https://pith.science/api/pith-number/TLXSX7XH5FPMKVUO4LIW4JLT3T/graph.json","fetch_events":"https://pith.science/api/pith-number/TLXSX7XH5FPMKVUO4LIW4JLT3T/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T/action/storage_attestation","attest_author":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T/action/author_attestation","sign_citation":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T/action/citation_signature","submit_replication":"https://pith.science/pith/TLXSX7XH5FPMKVUO4LIW4JLT3T/action/replication_record"}},"created_at":"2026-05-18T00:14:30.246458+00:00","updated_at":"2026-05-18T00:14:30.246458+00:00"}