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The resulting bound is uniform in both coefficients $c_{\\mathfrak{m}}$ and $N_i$'s. To prove this, we develop a higher-dimensional refinement of the multi-parameter circle method introduced in \\cite{B4,KS}, thereby"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2607.25955","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CA","submitted_at":"2026-07-28T16:41:33Z","cross_cats_sorted":[],"title_canon_sha256":"40681fd8573071f44eeda79594983b234987bfaaba22223e9d4e148f0b0b3833","abstract_canon_sha256":"4e7e64e813ec458927d0d75d0abd1d4f7237fb9d6ab12ed3249a639f8902d36e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-29T01:26:12.963740Z","signature_b64":"SmU36PeFDENadsOp0n9k9F8xPLuIoiP/u5afwoCRd1jyIjhKgU6FOiUz4dkcu/zFDnXzqFVkaOb8JQrMP6nzBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"844e3f51bf300413289ec878c64d406fe938aebad731a002bac1360c6b03e1c9","last_reissued_at":"2026-07-29T01:26:12.962881Z","signature_status":"signed_v1","first_computed_at":"2026-07-29T01:26:12.962881Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multi-Parameter Exponential Sums with Product Hilbert Kernels","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CA","authors_text":"Hoyoung Song, Joonil Kim","submitted_at":"2026-07-28T16:41:33Z","abstract_excerpt":"We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter singular exponential sum $$ \\sum_{|t_1|\\le N_1,\\dots,|t_k|\\le N_k} \\frac{e^{2\\pi i P(t_1,\\dots,t_k)}}{t_1\\cdots t_k}, $$ where $P:\\mathbb{Z}^k\\to\\mathbb{R}$ is a polynomial of the form $\nP(t)=\\sum_{\\mathfrak{m}\\in \\Lambda} c_{\\mathfrak{m}}\\, t^{\\mathfrak{m}}, $\nwith real coefficients. The resulting bound is uniform in both coefficients $c_{\\mathfrak{m}}$ and $N_i$'s. To prove this, we develop a higher-dimensional refinement of the multi-parameter circle method introduced in \\cite{B4,KS}, thereby"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.25955","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2607.25955/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2607.25955","created_at":"2026-07-29T01:26:12.963298+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.25955v1","created_at":"2026-07-29T01:26:12.963298+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.25955","created_at":"2026-07-29T01:26:12.963298+00:00"},{"alias_kind":"pith_short_12","alias_value":"QRHD6UN7GACB","created_at":"2026-07-29T01:26:12.963298+00:00"},{"alias_kind":"pith_short_16","alias_value":"QRHD6UN7GACBGKE6","created_at":"2026-07-29T01:26:12.963298+00:00"},{"alias_kind":"pith_short_8","alias_value":"QRHD6UN7","created_at":"2026-07-29T01:26:12.963298+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7","json":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7.json","graph_json":"https://pith.science/api/pith-number/QRHD6UN7GACBGKE6ZB4MMTKAN7/graph.json","events_json":"https://pith.science/api/pith-number/QRHD6UN7GACBGKE6ZB4MMTKAN7/events.json","paper":"https://pith.science/paper/QRHD6UN7"},"agent_actions":{"view_html":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7","download_json":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7.json","view_paper":"https://pith.science/paper/QRHD6UN7","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.25955&json=true","fetch_graph":"https://pith.science/api/pith-number/QRHD6UN7GACBGKE6ZB4MMTKAN7/graph.json","fetch_events":"https://pith.science/api/pith-number/QRHD6UN7GACBGKE6ZB4MMTKAN7/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7/action/timestamp_anchor","attest_storage":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7/action/storage_attestation","attest_author":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7/action/author_attestation","sign_citation":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7/action/citation_signature","submit_replication":"https://pith.science/pith/QRHD6UN7GACBGKE6ZB4MMTKAN7/action/replication_record"}},"created_at":"2026-07-29T01:26:12.963298+00:00","updated_at":"2026-07-29T01:26:12.963298+00:00"}