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Parshall","submitted_at":"2018-02-22T20:49:40Z","abstract_excerpt":"Let ${\\Bbb F}_q$ be a finite field of order $q.$ We prove that if $d\\ge 2$ is even and $E \\subset {\\Bbb F}_q^d$ with $|E| \\ge 9q^{\\frac{d}{2}}$ then $$ {\\Bbb F}_q=\\frac{\\Delta(E)}{\\Delta(E)}=\\left\\{ \\frac{a}{b}: a \\in \\Delta(E), b \\in \\Delta(E) \\backslash \\{0\\} \\right\\},$$ where $$ \\Delta(E)=\\{||x-y||: x,y \\in E\\}, \\ ||x||=x_1^2+x_2^2+\\cdots+x_d^2.$$ If the dimension $d$ is odd and $E\\subset \\mathbb F_q^d$ with $|E|\\ge 6q^{\\frac{d}{2}},$ then $$ \\{0\\}\\cup\\mathbb F_q^+ \\subset \\frac{\\Delta(E)}{\\Delta(E)},$$ where $\\mathbb F_q^+$ denotes the set of nonzero quadratic residues in $\\mathbb F_q.$ Bo"},"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":"1802.08297","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CA","submitted_at":"2018-02-22T20:49:40Z","cross_cats_sorted":["math.CO","math.NT"],"title_canon_sha256":"bf331b493c4c83835b61d0ad1176e73b4e01303efe91b8b8e05f477cdcb394a4","abstract_canon_sha256":"bd0406ea0f56d3345947b725697a297b880d92702f4e063ef681c8e775c0356e"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:44:59.805441Z","signature_b64":"4uo+IHGFwWYKDQxq4380H9Fe0JFIXwmlopU+xXxg2y0sr1wht++y3xaBPffvYi20CJZWJu8cXeEyw97FqPsrBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0c050be375e7edc0ece633226a486f5a60f5e0f4b8982f1a15e049b082e597a2","last_reissued_at":"2026-05-17T23:44:59.804785Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:44:59.804785Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the quotient set of the distance set","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.NT"],"primary_cat":"math.CA","authors_text":"A. 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Parshall","submitted_at":"2018-02-22T20:49:40Z","abstract_excerpt":"Let ${\\Bbb F}_q$ be a finite field of order $q.$ We prove that if $d\\ge 2$ is even and $E \\subset {\\Bbb F}_q^d$ with $|E| \\ge 9q^{\\frac{d}{2}}$ then $$ {\\Bbb F}_q=\\frac{\\Delta(E)}{\\Delta(E)}=\\left\\{ \\frac{a}{b}: a \\in \\Delta(E), b \\in \\Delta(E) \\backslash \\{0\\} \\right\\},$$ where $$ \\Delta(E)=\\{||x-y||: x,y \\in E\\}, \\ ||x||=x_1^2+x_2^2+\\cdots+x_d^2.$$ If the dimension $d$ is odd and $E\\subset \\mathbb F_q^d$ with $|E|\\ge 6q^{\\frac{d}{2}},$ then $$ \\{0\\}\\cup\\mathbb F_q^+ \\subset \\frac{\\Delta(E)}{\\Delta(E)},$$ where $\\mathbb F_q^+$ denotes the set of nonzero quadratic residues in $\\mathbb F_q.$ Bo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.08297","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":"1802.08297","created_at":"2026-05-17T23:44:59.804887+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.08297v1","created_at":"2026-05-17T23:44:59.804887+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.08297","created_at":"2026-05-17T23:44:59.804887+00:00"},{"alias_kind":"pith_short_12","alias_value":"BQCQXY3V47W4","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_16","alias_value":"BQCQXY3V47W4B3HG","created_at":"2026-05-18T12:32:16.446611+00:00"},{"alias_kind":"pith_short_8","alias_value":"BQCQXY3V","created_at":"2026-05-18T12:32:16.446611+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/BQCQXY3V47W4B3HGGMRGUSDPLJ","json":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ.json","graph_json":"https://pith.science/api/pith-number/BQCQXY3V47W4B3HGGMRGUSDPLJ/graph.json","events_json":"https://pith.science/api/pith-number/BQCQXY3V47W4B3HGGMRGUSDPLJ/events.json","paper":"https://pith.science/paper/BQCQXY3V"},"agent_actions":{"view_html":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ","download_json":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ.json","view_paper":"https://pith.science/paper/BQCQXY3V","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.08297&json=true","fetch_graph":"https://pith.science/api/pith-number/BQCQXY3V47W4B3HGGMRGUSDPLJ/graph.json","fetch_events":"https://pith.science/api/pith-number/BQCQXY3V47W4B3HGGMRGUSDPLJ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ/action/storage_attestation","attest_author":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ/action/author_attestation","sign_citation":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ/action/citation_signature","submit_replication":"https://pith.science/pith/BQCQXY3V47W4B3HGGMRGUSDPLJ/action/replication_record"}},"created_at":"2026-05-17T23:44:59.804887+00:00","updated_at":"2026-05-17T23:44:59.804887+00:00"}