{"paper":{"title":"Sharp Ternary Martingale Isoperimetry and $n$-adic Takagi-Type Lower Bounds","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.PR","authors_text":"Natanael Alpay","submitted_at":"2026-07-13T04:17:27Z","abstract_excerpt":"Let $S_1$ be the one-variation associated with the regular $n$-adic martingale filtration on $[0,1)$. We study the martingale isoperimetric profile \\[ V_n(x):= \\inf_{\\substack{A\\subset[0,1)\\ {\\rm measurable}\\\\ |A|=x}} \\|S_1(\\mathbbm 1_A)\\|_1 . \\] For the ternary filtration we determine this profile exactly. Namely, \\[ V_3(x)=T_3(x):= \\sum_{j=0}^{\\infty}3^{-j}\\psi_3(\\{3^j x\\}), \\] where \\[ \\psi_3(t)= \\min\\left\\{ \\frac{1+2\\left|t-\\frac12\\right|}{3}, \\frac{2-4\\left|t-\\frac12\\right|}{3} \\right\\}, \\qquad 0\\le t\\le1 . \\] Thus the sharp ternary profile is a Takagi-type Bellman function. It is, howeve"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.11069","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.11069/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"}