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Almost surely on non-extinction, $\\dim_F(\\mu)=\\dim_E(\\mu)=\\dim_2(\\mu)=\\sup_{1<q<2}\\max\\{0,2-(2/q)(1+\\log_2\\mathbb{E}[W^q])\\}$, with the convention that the corresponding term is zero when $\\mathbb{E}[W^q]=\\infty$. The proof is carried out in a vector-valued cascade model allowing arbitrary dependence between sibling weights; the classical independent cascade is"},"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":"2606.08683","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2026-06-07T15:35:06Z","cross_cats_sorted":[],"title_canon_sha256":"5502923ff7a149b927de0a819ec51b40e93bda3a99628b894201d95843c0cd89","abstract_canon_sha256":"c3175c234df58c843b5e627d954d8fc82349b58650ca24c1559278c2a601beae"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-09T01:05:47.362407Z","signature_b64":"nS+XHrwm/vh3r1hL+P68vSewTOriA0/xzMqtvnjUXEm4bzcVDVu94mZyuWzRot0ZY8p1e/byzxN5OJIVCfM8Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"38de39b67323a7f0a719477a7a58dbfb859f8ccb71dee0f5012b27b0a706ff5e","last_reissued_at":"2026-06-09T01:05:47.361937Z","signature_status":"signed_v1","first_computed_at":"2026-06-09T01:05:47.361937Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Exact Fourier dimensions of dyadic Mandelbrot cascades under minimal integrability","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Chengbo Xiao, Guozheng Cheng, Hongdou Qu, Menghan Li, Xiang Fang, Yin Cai","submitted_at":"2026-06-07T15:35:06Z","abstract_excerpt":"We determine the Fourier dimension of the canonical dyadic Mandelbrot cascade on the unit interval under the minimal Kahane--Peyriere condition $W \\ge 0$, $\\mathbb{E}W=1$, $\\mathbb{E}[W\\log_2^+ W]<\\infty$, and $\\mathbb{E}[W\\log_2 W]<1$. Almost surely on non-extinction, $\\dim_F(\\mu)=\\dim_E(\\mu)=\\dim_2(\\mu)=\\sup_{1<q<2}\\max\\{0,2-(2/q)(1+\\log_2\\mathbb{E}[W^q])\\}$, with the convention that the corresponding term is zero when $\\mathbb{E}[W^q]=\\infty$. 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