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For $k \\ge 4$, we bound the number of variables needed to ensure that if $\\eta$ is real and $\\tau > 0$ is sufficiently large then there exist integers $x_1 > \\mu_1, \\ldots, x_s > \\mu_s$ such that $|\\mathfrak{F}(\\mathbf{x}) - \\tau| < \\eta$. This is a real analogue to Waring's problem. When $s \\ge 2k^2-2k+3$, we provide an asymptotic formula. We prove similar results for sums of general univariate degree $k"},"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":"1409.4259","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-09-12T16:43:35Z","cross_cats_sorted":[],"title_canon_sha256":"566ef77f0b23d29a8ebea729e2e0132bbd604bb827d1d5cf6754ea54088e7469","abstract_canon_sha256":"1ae6726caeabad75ea5ae03a65e39b81ce4f275e405d0c8386c5f7df9a927dbe"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:25:03.069584Z","signature_b64":"xuW3CYHL7TAkMxYfJ10Fer7KsnXW7C+MeZgN+XDY6sHkrq9NXYCHeNsxioSAklePjNCHovVB3yDb1Y/3TrbdAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"50a3aa4fc9b40ae59020694b82c104cf2e5dc657c6835c8279a6f590f26d0871","last_reissued_at":"2026-05-18T01:25:03.068966Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:25:03.068966Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Waring's problem with shifts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Sam Chow","submitted_at":"2014-09-12T16:43:35Z","abstract_excerpt":"Let $\\mu_1, \\ldots, \\mu_s$ be real numbers, with $\\mu_1$ irrational. We investigate sums of shifted $k$th powers $\\mathfrak{F}(x_1, \\ldots, x_s) = (x_1 - \\mu_1)^k + \\ldots + (x_s - \\mu_s)^k$. For $k \\ge 4$, we bound the number of variables needed to ensure that if $\\eta$ is real and $\\tau > 0$ is sufficiently large then there exist integers $x_1 > \\mu_1, \\ldots, x_s > \\mu_s$ such that $|\\mathfrak{F}(\\mathbf{x}) - \\tau| < \\eta$. This is a real analogue to Waring's problem. When $s \\ge 2k^2-2k+3$, we provide an asymptotic formula. 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