{"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. Given nine variables"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.4258","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"}