{"paper":{"title":"Sets of bounded remainder for the continuous irrational rotation on $[0,1)^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gerhard Larcher, Sigrid Grepstad","submitted_at":"2016-03-01T10:02:55Z","abstract_excerpt":"We study sets of bounded remainder for the two-dimensional continuous irrational rotation $(\\{x_1+t\\}, \\{x_2+t\\alpha \\})_{t \\geq 0}$ in the unit square. In particular, we show that for almost all $\\alpha$ and every starting point $(x_1, x_2)$, every polygon $S$ with no edge of slope $\\alpha$ is a set of bounded remainder. Moreover, every convex set $S$ whose boundary is twice continuously differentiable with positive curvature at every point is a bounded remainder set for almost all $\\alpha$ and every starting point $(x_1, x_2)$. Finally we show that these assertions are, in some sense, best p"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.00207","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"}