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We prove that the occupation measure for $f$ has $L^2$ density almost surely. Furthermore, the Hausdorff dimension of the graph of $f$ is almost surely equal to $D = 2+ \\log{a}/\\log{b}$ provided $ b = \\lim_{n\\rightarrow \\infty}b_{n+1}/b_n>1$ and $ab>1$."},"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":"1312.2812","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.DS","submitted_at":"2013-12-10T14:31:22Z","cross_cats_sorted":[],"title_canon_sha256":"2dae16f4a6b0f3cff1481cf01bc382d8d2e99ec747a003faee353b8f0e87d832","abstract_canon_sha256":"58aab449b0f6a681629a3692d6edb8a5e164728bb392eec312ec93af5b2cd6d6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:45:52.130182Z","signature_b64":"jKA78h0ShTKLlDmRuD2zwUJ1gE9A6X6NPY2+REgPoUH49vurRebVtXhvltNnA9fQcf5R3+MMqvzq3DQsk7wgCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b30f164bdfcddee26e64f7a4bd7fd364d8b4631710720c79e8c7b4e23ffa6e53","last_reissued_at":"2026-05-18T01:45:52.129688Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:45:52.129688Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Measure and Hausdorff dimension of randomized Weierstrass-type functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.DS","authors_text":"Julia Romanowska","submitted_at":"2013-12-10T14:31:22Z","abstract_excerpt":"In this paper we consider functions of the type $$f(x) = \\sum_{n=0}^\\infty a_n g(b_nx+\\theta_n),$$ where $(a_n)$ are independent random variables uniformly distributed on $(-a^n, a^n)$ for some $0<a<1$, $b_{n+1}/b_n \\geq b >1$, $a^2b> 1$ and $g$ is a $C^1$ periodic real function with finite number of critical points in every bounded interval. We prove that the occupation measure for $f$ has $L^2$ density almost surely. Furthermore, the Hausdorff dimension of the graph of $f$ is almost surely equal to $D = 2+ \\log{a}/\\log{b}$ provided $ b = \\lim_{n\\rightarrow \\infty}b_{n+1}/b_n>1$ and $ab>1$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.2812","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":"1312.2812","created_at":"2026-05-18T01:45:52.129756+00:00"},{"alias_kind":"arxiv_version","alias_value":"1312.2812v1","created_at":"2026-05-18T01:45:52.129756+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1312.2812","created_at":"2026-05-18T01:45:52.129756+00:00"},{"alias_kind":"pith_short_12","alias_value":"WMHRMS67ZXPO","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_16","alias_value":"WMHRMS67ZXPOE3TE","created_at":"2026-05-18T12:28:04.890932+00:00"},{"alias_kind":"pith_short_8","alias_value":"WMHRMS67","created_at":"2026-05-18T12:28:04.890932+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/WMHRMS67ZXPOE3TE66SL276TMT","json":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT.json","graph_json":"https://pith.science/api/pith-number/WMHRMS67ZXPOE3TE66SL276TMT/graph.json","events_json":"https://pith.science/api/pith-number/WMHRMS67ZXPOE3TE66SL276TMT/events.json","paper":"https://pith.science/paper/WMHRMS67"},"agent_actions":{"view_html":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT","download_json":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT.json","view_paper":"https://pith.science/paper/WMHRMS67","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1312.2812&json=true","fetch_graph":"https://pith.science/api/pith-number/WMHRMS67ZXPOE3TE66SL276TMT/graph.json","fetch_events":"https://pith.science/api/pith-number/WMHRMS67ZXPOE3TE66SL276TMT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT/action/storage_attestation","attest_author":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT/action/author_attestation","sign_citation":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT/action/citation_signature","submit_replication":"https://pith.science/pith/WMHRMS67ZXPOE3TE66SL276TMT/action/replication_record"}},"created_at":"2026-05-18T01:45:52.129756+00:00","updated_at":"2026-05-18T01:45:52.129756+00:00"}