{"paper":{"title":"On the Hausdorff dimension of graph of random vector-valued Weierstrass function","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The Hausdorff dimension of the graph of the random vector-valued Weierstrass function equals 3-2β with probability one.","cross_cats":["math.PR"],"primary_cat":"math.CA","authors_text":"Jun Jason Luo, Zi-Rui Zhang","submitted_at":"2026-04-15T14:20:03Z","abstract_excerpt":"Let $\\Theta=\\{\\theta_n\\}, \\Lambda=\\{\\lambda_n\\}$ be two sequences of independent and identically distributed uniform random variables on $[0,1]$. The random vector-valued Weierstrass function is given by\n  $$\n  f_{\\Theta,\\Lambda}(x)=\n  \\left(\n  \\sum_{n=0}^{\\infty} a^n\\cos\\bigl(2\\pi (b^n x+\\theta_n)\\bigr),\\\n  \\sum_{n=0}^{\\infty} a^n\\sin\\bigl(2\\pi (b^n x+\\lambda_n)\\bigr)\n  \\right), \\; x\\in[0,1],\n  $$ where $0<a<1<b,\\ ab> 1$. The Hausdorff dimension of the graph of this function is proved to be\n  $$\\dim_H G(f_{\\Theta,\\Lambda}) = \\min\\left\\{-\\frac{\\log b}{\\log a}, \\, 3 +2\\frac{\\log a}{\\log b}\\righ"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We prove that, with probability one, the Hausdorff dimension of the graph of this function is dim_H G(f_Θ,Λ)=3-2β, extending a result of Hunt in 1998.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The two sequences of phases Θ and Λ consist of independent and identically distributed uniform random variables on [0,1], with the contraction parameter β strictly less than 1/2.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"With probability one, the Hausdorff dimension of the graph of the random vector-valued Weierstrass function is 3-2β.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The Hausdorff dimension of the graph of the random vector-valued Weierstrass function equals 3-2β with probability one.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"83c916bac91ff131829cbdf7dc903b3d14f4d2cd5bd34b31e8e4fde7bdb59f8c"},"source":{"id":"2604.13913","kind":"arxiv","version":2},"verdict":{"id":"67d38e1b-f28e-49c7-82fd-c1508e47acca","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T11:45:03.814014Z","strongest_claim":"We prove that, with probability one, the Hausdorff dimension of the graph of this function is dim_H G(f_Θ,Λ)=3-2β, extending a result of Hunt in 1998.","one_line_summary":"With probability one, the Hausdorff dimension of the graph of the random vector-valued Weierstrass function is 3-2β.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The two sequences of phases Θ and Λ consist of independent and identically distributed uniform random variables on [0,1], with the contraction parameter β strictly less than 1/2.","pith_extraction_headline":"The Hausdorff dimension of the graph of the random vector-valued Weierstrass function equals 3-2β with probability one."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.13913/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}