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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("},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1504.03251","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-04-02T10:14:21Z","cross_cats_sorted":[],"title_canon_sha256":"c200277a38128c48bbc6a478aeb661999f89fe6622340d835054dd271d7059a3","abstract_canon_sha256":"6344b40766619530abff980cdff5b76dd7806b20e2a44def23d6029ebd485811"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:18:58.813894Z","signature_b64":"pASSkr6y22rM49iFSg/da8/JwIrVRbRYjsdISOkzUK9qT5KKXZWZCN0MkT23woi6ogA8JIDFTiSTzWKCHwGIDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"02a3e6b0196f5c6f7cd77baf7cbf3a6904b97871f460fd31caab0747c725fc56","last_reissued_at":"2026-05-18T02:18:58.813271Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:18:58.813271Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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 $\\mathbb{R}^{d}$. 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