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Hlawka showed that $W_{n}(D)\\le 2^{n}$, where $D$ is an origin-symmetric convex body.\n  We sharpen Hlawka's estimates for $D$ being the ball $B^{n}$ and the cube $I^{n}$. In particular, we prove that $W_{n}(B^{n})\\le 2^{(0.401\\ldots +o(1))n}$. We also obtain a lower bound of $W_{n}(D)$. Moreover, for a cube $ D=\\fra"},"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":"1604.01302","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2016-04-05T15:36:24Z","cross_cats_sorted":[],"title_canon_sha256":"4de57fbc822ebec3cce4e82be20c81f1f718aabf353c73b901a0fb73cb685332","abstract_canon_sha256":"308eb7dd1c2f02e78831ee9923c566931192c628fefb02301e677d058aeb79a6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:17:38.556913Z","signature_b64":"dPJczMATgshRVCfJIhEaOfh8iOiiUPbo7aPoTu6ZDtxy+rRs0VE9q+/Y+Zm/sq1WwCluEkMfHybvO/ZsNQotAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"386855f0bed15211d75eaeb67be819d48743a7a0dedfca318b0d7ac0b4680aad","last_reissued_at":"2026-05-18T01:17:38.556198Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:17:38.556198Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Wiener's problem for positive definite functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Dmitry Gorbachev, Sergey Tikhonov","submitted_at":"2016-04-05T15:36:24Z","abstract_excerpt":"We study the sharp constant $W_{n}(D)$ in Wiener's inequality for positive definite functions \\[ \\int_{\\mathbb{T}^{n}}|f|^{2}\\,dx\\le W_{n}(D)|D|^{-1}\\int_{D}|f|^{2}\\,dx,\\quad D\\subset \\mathbb{T}^{n}. \\] N. Wiener proved that $W_{1}([-\\delta,\\delta])<\\infty$, $\\delta\\in (0,1/2)$. E. Hlawka showed that $W_{n}(D)\\le 2^{n}$, where $D$ is an origin-symmetric convex body.\n  We sharpen Hlawka's estimates for $D$ being the ball $B^{n}$ and the cube $I^{n}$. In particular, we prove that $W_{n}(B^{n})\\le 2^{(0.401\\ldots +o(1))n}$. We also obtain a lower bound of $W_{n}(D)$. 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