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As a consequence, it is shown that if $E\\subset \\mathbb F_q^d$ with $|E|\\geq C q^{\\frac{d+1}{2}-\\frac{1}{6d+2}},$ then $|\\Delta_3(E)|\\g"},"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":"1403.6138","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-03-24T20:22:33Z","cross_cats_sorted":["math.CA","math.NT"],"title_canon_sha256":"a06994d83f5df2d65ee1a2d78a6b18a3c63c05d8d9a95b41413f5aaf519bdb96","abstract_canon_sha256":"390ce179b4849f3c53f93e86edf340657120e035e2fe6bea93bc011d86dedb8d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:57.196364Z","signature_b64":"n9UQM2moxgg7TEqgIlx/l3B38rXYPpFrRDjU+xG2Uee9qg+xCAKhOXBoeYHh5A1TwQ7zH1gioHntSgW+HXNNAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0b83696ff39e7f97eee756d56cc042b769f476e9d6b81d4eea18993c18465e36","last_reissued_at":"2026-05-18T02:27:57.195623Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:57.195623Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the sums of any k points in finite fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.CO","authors_text":"David Covert, Doowon Koh, Youngjin Pi","submitted_at":"2014-03-24T20:22:33Z","abstract_excerpt":"For a set $E\\subset \\mathbb F_q^d$, we define the $k$-resultant magnitude set as $ \\Delta_k(E) =\\{\\|\\textbf{x}_1 + \\dots + \\textbf{x}_k\\|\\in \\mathbb F_q: \\textbf{x}_1, \\dots, \\textbf{x}_k \\in E\\},$ where $\\|\\textbf{v}\\|=v_1^2+\\cdots+ v_d^2$ for $\\textbf{v}=(v_1, \\ldots, v_d) \\in \\mathbb F_q^d.$ In this paper we find a connection between a lower bound of the cardinality of the $k$-resultant magnitude set and the restriction theorem for spheres in finite fields. 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