{"paper":{"title":"A discrete harmonic function bounded on a large portion of $\\mathbb{Z}^2$ is constant","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.CA","authors_text":"Alexander Logunov, Eugenia Malinnikova, Lev Buhovsky, Mikhail Sodin","submitted_at":"2017-12-21T12:26:51Z","abstract_excerpt":"An improvement of the Liouville theorem for discrete harmonic functions on $\\mathbb{Z}^2$ is obtained. More precisely, we prove that there exists a positive constant $\\varepsilon$ such that if $u$ is discrete harmonic on $\\mathbb{Z}^2$ and for each sufficiently large square $Q$ centered at the origin $|u|\\le 1$ on a $(1-\\varepsilon)$ portion of $Q$ then $u$ is constant."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.07902","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":""},"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"}