{"paper":{"title":"Additive equations in dense variables via truncated restriction estimates","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Kevin Henriot","submitted_at":"2015-08-24T19:34:39Z","abstract_excerpt":"We study translation-invariant additive equations of the form $\\sum_{i=1}^s \\lambda_i \\mathbf{P}(\\mathbf{n}_i) = 0$ in variables $\\mathbf{n}_i \\in \\mathbb{Z}^d$, where the $\\lambda_i$ are nonzero integers summing to zero, and $\\mathbf{P}$ is a system of homogeneous polynomials such that the above equation is invariant by translation. We investigate the solvability of this equation in subsets of density $(\\log N)^{-c(\\mathbf{P},\\mathbf{\\lambda})}$ of a large box $[N]^d$, via the energy increment method. We obtain positive results in roughly the number of variables currently needed to derive a c"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.05923","kind":"arxiv","version":5},"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"}