{"paper":{"title":"On Vu's restricted box estimate in Waring's problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"The restricted box estimate in Waring's problem holds with power-saving error for s at least k squared minus k plus order square root of k, improving on the prior exponential threshold.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Christian T\\'afula","submitted_at":"2026-05-14T16:57:00Z","abstract_excerpt":"In 2000, Vu proved that the number of solutions of $x_1^k + \\cdots + x_s^k = N$ in an arbitrary box satisfies the expected Hardy--Littlewood upper bound with a power-saving error term, for $s \\geq O(8^k k^3)$. We show that one may take $s\\geq k^2 - k + O(\\sqrt{k})$."},"claims":{"count":3,"items":[{"kind":"strongest_claim","text":"We show that one may take s ≥ k² - k + O(√k).","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The minor-arc estimates or box-restriction handling in the circle method continue to deliver a power-saving error once s reaches the new quadratic threshold; this is not verified from the abstract alone.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The restricted box estimate in Waring's problem holds with power-saving error for s at least k squared minus k plus order square root of k, improving on the prior exponential threshold.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"}],"snapshot_sha256":"109d41c00341564e7c44effd4c3df43df4669e797b2902750d1ef288d976f9e7"},"source":{"id":"2605.15067","kind":"arxiv","version":1},"verdict":{"id":"e50f3cd3-f7f2-4ea9-aa0f-c405687d7662","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-15T03:20:45.546698Z","strongest_claim":"We show that one may take s ≥ k² - k + O(√k).","one_line_summary":"The restricted box estimate in Waring's problem holds with power-saving error for s at least k squared minus k plus order square root of k, improving on the prior exponential threshold.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The minor-arc estimates or box-restriction handling in the circle method continue to deliver a power-saving error once s reaches the new quadratic threshold; this is not verified from the abstract alone.","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"}