{"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. We prove similar results for sums of general univariate degree $k"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.4259","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"}