{"paper":{"title":"On the logarithmic equilibrium measure on curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Damian D\\k{a}browski, Tuomas Orponen","submitted_at":"2025-06-09T13:34:55Z","abstract_excerpt":"Let $\\mu$ be the logarithmic equilibrium measure on a compact set $\\gamma \\subset \\mathbb{R}^{d}$. We prove that $\\mu$ is absolutely continuous with respect to the length measure on the part of $\\gamma$ which can be locally expressed as the graph of a $C^{1,\\alpha}$-function $\\mathbb{R} \\to \\mathbb{R}^{d - 1}$, $\\alpha > 0$.\n  For $d = 2$, at least in the case where $\\gamma$ is a compact $C^{1,\\alpha}$-graph, our result can also be deduced from the classical fact that $\\mu$ coincides with the harmonic measure of $\\Omega =\\mathbb{R}^{2} \\, \\setminus \\, \\gamma$ with pole at $\\infty$. For $d \\geq"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.07752","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/2506.07752/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"}