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By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\\le r\\le k-1$. In particular, when $s,k\\in \\mathbb N$ satisfy $k\\ge 3$ and $1\\le s\\le (k^2-1)/2$, we establish the essentially diagonal behaviour $I_{s,k,1}(X)\\ll X^{s+\\epsilon}$."},"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":"1707.06047","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-19T12:37:13Z","cross_cats_sorted":[],"title_canon_sha256":"fa1736c1077749dd5152134e0a364b2c988d6900831767f0ab1c410537a6efe8","abstract_canon_sha256":"dbfe5b00111b25260bff09a85aa4074c97a617d9e7498e7c6e7b419b49925d7c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:39:57.722042Z","signature_b64":"KxSSPljYwUpNjmxpRJbtLjyn8CsnzAtjlJzfxL7GBPrITUCSnzjZicyLHyfey0XnNeKRo9j7O2W4k0SRgZCkCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5d7642ef643cd9654d7fd036c399c6307823e1a5cd84a42da2017c15ab463eac","last_reissued_at":"2026-05-18T00:39:57.721567Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:39:57.721567Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Vinogradov systems with a slice off","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Julia Brandes, Trevor D. Wooley","submitted_at":"2017-07-19T12:37:13Z","abstract_excerpt":"Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\\ldots +x_s^j=y_1^j+\\ldots +y_s^j\\quad (\\text{$1\\le j\\le k$, $j\\ne r$}),$$ with $1\\le x_i,y_i\\le X$ $(1\\le i\\le s)$. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\\le r\\le k-1$. In particular, when $s,k\\in \\mathbb N$ satisfy $k\\ge 3$ and $1\\le s\\le (k^2-1)/2$, we establish the essentially diagonal behaviour $I_{s,k,1}(X)\\ll X^{s+\\epsilon}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.06047","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":""},"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":"1707.06047","created_at":"2026-05-18T00:39:57.721635+00:00"},{"alias_kind":"arxiv_version","alias_value":"1707.06047v1","created_at":"2026-05-18T00:39:57.721635+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.06047","created_at":"2026-05-18T00:39:57.721635+00:00"},{"alias_kind":"pith_short_12","alias_value":"LV3EF33EHTMW","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_16","alias_value":"LV3EF33EHTMWKTL7","created_at":"2026-05-18T12:31:28.150371+00:00"},{"alias_kind":"pith_short_8","alias_value":"LV3EF33E","created_at":"2026-05-18T12:31:28.150371+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/LV3EF33EHTMWKTL72A3MHGOGGB","json":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB.json","graph_json":"https://pith.science/api/pith-number/LV3EF33EHTMWKTL72A3MHGOGGB/graph.json","events_json":"https://pith.science/api/pith-number/LV3EF33EHTMWKTL72A3MHGOGGB/events.json","paper":"https://pith.science/paper/LV3EF33E"},"agent_actions":{"view_html":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB","download_json":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB.json","view_paper":"https://pith.science/paper/LV3EF33E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1707.06047&json=true","fetch_graph":"https://pith.science/api/pith-number/LV3EF33EHTMWKTL72A3MHGOGGB/graph.json","fetch_events":"https://pith.science/api/pith-number/LV3EF33EHTMWKTL72A3MHGOGGB/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB/action/timestamp_anchor","attest_storage":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB/action/storage_attestation","attest_author":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB/action/author_attestation","sign_citation":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB/action/citation_signature","submit_replication":"https://pith.science/pith/LV3EF33EHTMWKTL72A3MHGOGGB/action/replication_record"}},"created_at":"2026-05-18T00:39:57.721635+00:00","updated_at":"2026-05-18T00:39:57.721635+00:00"}