{"paper":{"title":"Counting solutions to the quadratic determinant equation","license":"http://creativecommons.org/licenses/by/4.0/","headline":"When h equals N squared plus O(N), the number of solutions to x1 x2 minus x3 x4 equals h inside the box of side 2N admits an asymptotic with square-root cancellation error terms.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Akshat Mudgal, Jonathan Chapman","submitted_at":"2026-05-14T21:26:51Z","abstract_excerpt":"Given $h, N \\in \\mathbb{N}$ satisfying $1 \\leqslant h \\leqslant N^2$, we prove an asymptotic formula for the number of solutions to the equation $x_1 x_2 - x_3 x_4 = h$ with $x_1, \\ldots, x_4 \\in [-N,N] \\cap \\mathbb{Z}$. We use a combination of combinatorial and analytic arguments in physical space along with bounds for Kloosterman sums. Our main result concerns the case when $h = N^2 + O(N)$, wherein we obtain square-root cancellation error terms by bypassing Kloosterman sum bounds and exploiting an additional symmetry available in this setting via Ramanujan sums. This confirms a speculation "},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"When h = N^2 + O(N), the number of solutions admits an asymptotic formula with square-root cancellation error terms obtained by exploiting symmetry via Ramanujan sums and bypassing Kloosterman sum bounds.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The additional symmetry present precisely when h = N^2 + O(N) permits direct use of Ramanujan sums to achieve square-root cancellation without relying on general Kloosterman bounds.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"Proves asymptotic count of solutions to x1 x2 - x3 x4 = h for xi in [-N, N] with square-root cancellation when h = N^2 + O(N), confirming a prior speculation.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"When h equals N squared plus O(N), the number of solutions to x1 x2 minus x3 x4 equals h inside the box of side 2N admits an asymptotic with square-root cancellation error terms.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"65f5b80df6ff5052d89ea8b2264b67de2574bb67015bc82d172e7f2df7394d8d"},"source":{"id":"2605.15434","kind":"arxiv","version":1},"verdict":{"id":"08e2bdf6-f0da-4c2d-9e31-0e01c5381e8a","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-19T14:50:10.215864Z","strongest_claim":"When h = N^2 + O(N), the number of solutions admits an asymptotic formula with square-root cancellation error terms obtained by exploiting symmetry via Ramanujan sums and bypassing Kloosterman sum bounds.","one_line_summary":"Proves asymptotic count of solutions to x1 x2 - x3 x4 = h for xi in [-N, N] with square-root cancellation when h = N^2 + O(N), confirming a prior speculation.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The additional symmetry present precisely when h = N^2 + O(N) permits direct use of Ramanujan sums to achieve square-root cancellation without relying on general Kloosterman bounds.","pith_extraction_headline":"When h equals N squared plus O(N), the number of solutions to x1 x2 minus x3 x4 equals h inside the box of side 2N admits an asymptotic with square-root cancellation error terms."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2605.15434/integrity.json","findings":[],"available":true,"detectors_run":[{"name":"cited_work_retraction","ran_at":"2026-05-19T15:54:28.079071Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"citation_quote_validity","ran_at":"2026-05-19T15:50:29.980697Z","status":"completed","version":"0.1.0","findings_count":0},{"name":"doi_compliance","ran_at":"2026-05-19T15:01:38.358032Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"doi_title_agreement","ran_at":"2026-05-19T15:01:17.677136Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"claim_evidence","ran_at":"2026-05-19T14:21:54.127326Z","status":"completed","version":"1.0.0","findings_count":0},{"name":"ai_meta_artifact","ran_at":"2026-05-19T13:33:22.692019Z","status":"skipped","version":"1.0.0","findings_count":0}],"snapshot_sha256":"52c17a2e40dbb3e1ce80b10560a6066a9d2639ad6d05ae6c183156f87c061f3b"},"references":{"count":18,"sample":[{"doi":"","year":2026,"title":"M. Afifurrahman,A uniform formula on the number of integer matrices with given determinant and height, J. Number Theory281(2026), 741–770","work_id":"32676eb2-ef2d-4bb2-8c0e-1df3a59f738b","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":1976,"title":"Apostol,Introduction to analytic number theory, Undergraduate Texts in Mathematics, Springer- Verlag, New York-Heidelberg, 1976","work_id":"7028696b-1644-4236-b71f-c50128b5412e","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"J. Chapman, A. Mudgal,On commuting integer matrices, arXiv:2504.15839, to appear in Trans. Amer. Math. Soc","work_id":"dd70991f-c80b-4726-95e7-26ab68e4d652","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":null,"title":"J. Chapman, A. 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