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The functions $\\psi_{P,Q}$ are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of $Q$-normal, $Q$-ratio normal, and $Q$-distribution normal numbers and their pairwise intersections for fully divergent $Q$ that are infinite in limit. We analyz"},"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":"1310.2377","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-10-09T07:22:32Z","cross_cats_sorted":[],"title_canon_sha256":"0e3dc12c6454947feaf2977b6bd0f97434bdb5c23eddf66e6007924dc5e2f495","abstract_canon_sha256":"27df3bc491867178aed2e730906d679e5bb361237a9ee0f2116cb626d14a252b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:28:05.508982Z","signature_b64":"rDBdw3USrcoHLaVguza0C6BskxwkOcf9Af84V5MkZEHPIKOWQcxh54XwdjVJjD0zTGAHmeQ0WLQIqakxpXfnBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ace240176170b07d54dc4b4f2bbb7799773f75c517e0c4e042aa6635c89ff46a","last_reissued_at":"2026-05-18T02:28:05.508560Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:28:05.508560Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Number theoretic applications of a class of Cantor series fractal functions,I","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bill Mance","submitted_at":"2013-10-09T07:22:32Z","abstract_excerpt":"Suppose that $(P,Q) \\in \\mathbb{N}_2^{\\mathbb{N}} \\times \\mathbb{N}_2^{\\mathbb{N}}$ and $x=E_0.E_1E_2\\cdots$ is the $P$-Cantor series expansion of $x \\in \\mathbb{R}$. We define $\\psi_{P,Q}(x):=\\sum_{n=1}^\\infty \\frac {\\min(E_n,q_n-1)} {q_1 \\cdots q_n}$. The functions $\\psi_{P,Q}$ are used to construct many pathological examples of normal numbers. These constructions are used to give the complete containment relation between the sets of $Q$-normal, $Q$-ratio normal, and $Q$-distribution normal numbers and their pairwise intersections for fully divergent $Q$ that are infinite in limit. 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