{"paper":{"title":"A characterization theorem for the $L^{2}$-discrepancy of integer points in dilated polygons","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Giancarlo Travaglini, Maria Rosaria Tupputi","submitted_at":"2015-04-02T10:14:21Z","abstract_excerpt":"Let $C$ be a convex $d$-dimensional body. If $\\rho$ is a large positive number, then the dilated body $\\rho C$ contains $\\rho^{d}\\left\\vert C\\right\\vert +\\mathcal{O}\\left( \\rho^{d-1}\\right) $ integer points, where $\\left\\vert C\\right\\vert $ denotes the volume of $C$. The above error estimate $\\mathcal{O}\\left( \\rho^{d-1}\\right) $ can be improved in several cases. We are interested in the $L^{2}$-discrepancy $D_{C}(\\rho)$ of a copy of $\\rho C$ thrown at random in $\\mathbb{R}^{d}$. More precisely, we consider \\[ D_{C}(\\rho):=\\left\\{ \\int_{\\mathbb{T}^{d}}\\int_{SO(d)}\\left\\vert \\textrm{card}\\left("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.03251","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"}