{"paper":{"title":"Logarithmic bounds for translation-invariant equations in squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Kevin Henriot","submitted_at":"2014-09-16T04:34:41Z","abstract_excerpt":"We show that the equation $\\lambda_1 n_1^2 + ... + \\lambda_s n_s^2 = 0$ admits non-trivial solutions in any subset of $[N]$ of density $(\\log N)^{-c_s}$, provided that $s \\geq 7$ and the coefficients $\\lambda_i$ sum to zero and satisfy certain sign conditions. This improves upon previous known density bounds of the form $(\\log\\log N)^{-c}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1409.4501","kind":"arxiv","version":3},"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"}