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We show that if $\\eta$ is real, $\\tau >0$ is sufficiently large, and $s \\ge 9$, then there exist integers $x_1 > \\mu_1, \\ldots, x_s > \\mu_s$ such that $|F(\\mathbf{x})- \\tau| < \\eta$. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for $s \\ge 5$. For $s \\ge 11$, we provide an asymptotic formula. If $s \\ge 6$ then $F(\\mathbf{Z}^s)$ is dense on the reals. Given nine variables"},"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":"1409.4258","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-12T16:30:16Z","cross_cats_sorted":[],"title_canon_sha256":"d29959529251d1e1b7e117116086eddf56d37e8b7834cd9504bed04f5cf666c5","abstract_canon_sha256":"a7cdc59c5a9754e50356134f14c0ce3a45c2359fb741c1fb683e6a2d6b813188"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:53:09.168552Z","signature_b64":"nzk2Cgz4RAjbDgDoLEgYlynRHTl1KTHOaOi/S8CsM/MC+CIFiLwAUfzl3lAzaI2OgILcTV9iOiPCpclH7cdNBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6c0199d6a9e1fd7ea47d7dd7fcdba213c030192f01b7eec4a36d3fd82e528a1b","last_reissued_at":"2026-05-18T01:53:09.167973Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:53:09.167973Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums of cubes with shifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sam Chow","submitted_at":"2014-09-12T16:30:16Z","abstract_excerpt":"Let $\\mu_1, \\ldots, \\mu_s$ be real numbers, with $\\mu_1$ irrational. We investigate sums of shifted cubes $F(x_1,\\ldots,x_s) = (x_1 - \\mu_1)^3 + \\ldots + (x_s - \\mu_s)^3$. We show that if $\\eta$ is real, $\\tau >0$ is sufficiently large, and $s \\ge 9$, then there exist integers $x_1 > \\mu_1, \\ldots, x_s > \\mu_s$ such that $|F(\\mathbf{x})- \\tau| < \\eta$. This is a real analogue to Waring's problem. We then prove a full density result of the same flavour for $s \\ge 5$. For $s \\ge 11$, we provide an asymptotic formula. If $s \\ge 6$ then $F(\\mathbf{Z}^s)$ is dense on the reals. 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