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For any sufficiently large prime $p$ and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of $G=(G_1,\\dots, G_n)$, we study various properties regarding the distribution of the vectors by fractional parts \\begin{equation*}\n  \\bigg(\\bigg\\{ \\frac{G_1(\\textbf{x})}{p}\\bigg\\},\\cdots,\\bigg\\{ \\frac{G_n(\\textbf{x})}{p}\\bigg\\}\\bigg)\\in \\mathbb{T}^n,\\hspace{10pt} \\textbf{x}\\in \\mathbb{F}_p^m. \\end{equation*} We prove refinemen"},"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":"2403.05078","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-03-08T06:13:32Z","cross_cats_sorted":[],"title_canon_sha256":"4bf2bb42e2e9b9a9fb42ec69200d5519ebe19163898f7fa4103ac86c2b74b143","abstract_canon_sha256":"742d2469d04522034c2dc294166b85aa8b40479f0f99e5a3ae9a9d2317f2ee31"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T01:04:48.987248Z","signature_b64":"uPAunXBtCcTACb1QvuoAUkJ5IxE8RIblGtGxju0zNogQSgZiq7CmSX4LIkI+iwlvQM8m6pv9Oxr1yjt8FiwcCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b74a07aa44dc78dc97c4e70ffcaf7b7361d9670b0da44eaed9893e7233d83c45","last_reissued_at":"2026-05-20T01:04:48.986483Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T01:04:48.986483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Distribution of solutions to systems of congruences in balls","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael Harm","submitted_at":"2024-03-08T06:13:32Z","abstract_excerpt":"Let $G_1,\\dots, G_n\\in \\mathbb{F}_p[X_1,\\dots,X_m]$ be $n$ polynomials in $m$ variables over the finite field $\\mathbb{F}_p$ of $p$ elements. For any sufficiently large prime $p$ and non-trivial bounds for the Weyl sums associated to the non-trivial linear combinations of $G=(G_1,\\dots, G_n)$, we study various properties regarding the distribution of the vectors by fractional parts \\begin{equation*}\n  \\bigg(\\bigg\\{ \\frac{G_1(\\textbf{x})}{p}\\bigg\\},\\cdots,\\bigg\\{ \\frac{G_n(\\textbf{x})}{p}\\bigg\\}\\bigg)\\in \\mathbb{T}^n,\\hspace{10pt} \\textbf{x}\\in \\mathbb{F}_p^m. \\end{equation*} We prove refinemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.05078","kind":"arxiv","version":2},"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/2403.05078/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":"2403.05078","created_at":"2026-05-20T01:04:48.986597+00:00"},{"alias_kind":"arxiv_version","alias_value":"2403.05078v2","created_at":"2026-05-20T01:04:48.986597+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2403.05078","created_at":"2026-05-20T01:04:48.986597+00:00"},{"alias_kind":"pith_short_12","alias_value":"W5FAPKSE3R4N","created_at":"2026-05-20T01:04:48.986597+00:00"},{"alias_kind":"pith_short_16","alias_value":"W5FAPKSE3R4NZF6E","created_at":"2026-05-20T01:04:48.986597+00:00"},{"alias_kind":"pith_short_8","alias_value":"W5FAPKSE","created_at":"2026-05-20T01:04:48.986597+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/W5FAPKSE3R4NZF6E44H7ZL33ON","json":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON.json","graph_json":"https://pith.science/api/pith-number/W5FAPKSE3R4NZF6E44H7ZL33ON/graph.json","events_json":"https://pith.science/api/pith-number/W5FAPKSE3R4NZF6E44H7ZL33ON/events.json","paper":"https://pith.science/paper/W5FAPKSE"},"agent_actions":{"view_html":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON","download_json":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON.json","view_paper":"https://pith.science/paper/W5FAPKSE","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2403.05078&json=true","fetch_graph":"https://pith.science/api/pith-number/W5FAPKSE3R4NZF6E44H7ZL33ON/graph.json","fetch_events":"https://pith.science/api/pith-number/W5FAPKSE3R4NZF6E44H7ZL33ON/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON/action/timestamp_anchor","attest_storage":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON/action/storage_attestation","attest_author":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON/action/author_attestation","sign_citation":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON/action/citation_signature","submit_replication":"https://pith.science/pith/W5FAPKSE3R4NZF6E44H7ZL33ON/action/replication_record"}},"created_at":"2026-05-20T01:04:48.986597+00:00","updated_at":"2026-05-20T01:04:48.986597+00:00"}