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This means that we can find a non-negative measurable \"witness function\"\n  $f: {\\mathbb R}\\to [0,+ {\\infty})$ such that both the convergence set $C(f, {\\Lambda})=\\{x: s(x)<+ {\\infty} \\}$ and its complement the divergence set $D(f, {\\Lambda})=\\{x: s(x)=+ {\\infty} \\}$ are of positive Lebesgue measure. If $ {\\Lambda}$ is not type $2$ we say that $ {\\Lambda}$ is type $1$.\n  The main result of our pa"},"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":"1804.10408","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-04-27T09:17:54Z","cross_cats_sorted":["math.PR"],"title_canon_sha256":"a946d21704728020a210953a58d90eca834a889bb5c16e5a5606a50372ad04c3","abstract_canon_sha256":"52f0a752e9d07d71922a0f177a8dfc8c725e6c5cc7f06961848739bdc60f13ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:17:20.873600Z","signature_b64":"E44NCG+NNa1Wr42bXQt/1nUfw4QUSDbmxgU+4vzYbb0pELjrxkCVpB8Wbb0FszlDQavpSZTXyY53GPZvBX4uCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c156ced38c6781ed7366125c39e69980c27ff6fe989c38bfcd4259e56acf1dd","last_reissued_at":"2026-05-18T00:17:20.872935Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:17:20.872935Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Random constructions for translates of non-negative functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Bal\\'azs Maga, Bruce Hanson, G\\'asp\\'ar V\\'ertesy, Zolt\\'an Buczolich","submitted_at":"2018-04-27T09:17:54Z","abstract_excerpt":"Suppose $\\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\\Lambda}$ is type $2$ if the series $s(x)=\\sum_{\\lambda\\in\\Lambda}f(x+\\lambda)$ does not satisfy a zero-one law. This means that we can find a non-negative measurable \"witness function\"\n  $f: {\\mathbb R}\\to [0,+ {\\infty})$ such that both the convergence set $C(f, {\\Lambda})=\\{x: s(x)<+ {\\infty} \\}$ and its complement the divergence set $D(f, {\\Lambda})=\\{x: s(x)=+ {\\infty} \\}$ are of positive Lebesgue measure. 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