{"paper":{"title":"Integral Representations and Asymptotics for a Family of Areal Mahler Measures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Quanyu Tang, Shu Zhang","submitted_at":"2026-08-03T09:23:17Z","abstract_excerpt":"We answer a problem posed by Matilde Lal\\'in concerning the areal Mahler measures of the multivariable polynomial family $$\n  P_n(x_1,\\ldots,x_n,u)\n  =\n  \\prod_{j=1}^n(1+x_j)\n  +\n  u\\prod_{j=1}^n(1-x_j),\n  \\qquad n\\geq1. $$ Using a probabilistic reformulation, we derive convolution and one-dimensional Fourier integral representations for $\\mathrm m_{\\mathbb D}(P_n)$. These representations yield strict alternating signs for all forward differences of the sequence $\\bigl(\\mathrm m_{\\mathbb D}(P_n)\\bigr)_{n\\geq1}$, as well as an explicit three-term asymptotic expansion as $n\\to\\infty$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.01951","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/2608.01951/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"}