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Let $r$ be a prime divisor of $q-1$ such that the largest prime power part of $q-1$ has the form $r^s$. Then there is a constant $0<\\epsilon<1$ such that for a ratio at least $ {q^{-\\epsilon h}}$ of $\\alpha\\in \\mathbb{F}_{q^{h}} \\backslash\\mathbb{F}_{q}$, the set $S=\\{ \\alpha-x^t, x\\in\\mathbb{F}_{q}\\}$ of cardinality $1+\\frac {q-1} {M(h)}$ contains a non-d-th power i"},"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":"1708.05976","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-08-20T14:43:26Z","cross_cats_sorted":["cs.CC","math.CO"],"title_canon_sha256":"4a2735aadcfce6c10711e48591ed62e13df77db0472a2d47831473ff6f32d702","abstract_canon_sha256":"632d18b773f2e47fbf5993a05476de8326a603214541181bdb22f0b3c1d6612b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:02.660511Z","signature_b64":"MmbhlYOK42Dd6YxJIdKmMNlqiiTs87HbIGRJPof23Mpg4tUt943ii8RH1Mc/cBtrOGWZ1D1SqsDM3peez767Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7d15152f0a368613ae2eead056e3422c0f1b349dd0605a4d6e3ab0b6ca2eb662","last_reissued_at":"2026-05-18T00:36:02.660146Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:02.660146Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the construction of small subsets containing special elements in a finite field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","math.CO"],"primary_cat":"math.NT","authors_text":"Jiyou Li","submitted_at":"2017-08-20T14:43:26Z","abstract_excerpt":"In this note we construct a series of small subsets containing a non-d-th power element in a finite field by applying certain bounds on incomplete character sums.\n  Precisely, let $h=\\lfloor q^{\\delta}\\rfloor>1$ and $d\\mid q^h-1$. Let $r$ be a prime divisor of $q-1$ such that the largest prime power part of $q-1$ has the form $r^s$. Then there is a constant $0<\\epsilon<1$ such that for a ratio at least $ {q^{-\\epsilon h}}$ of $\\alpha\\in \\mathbb{F}_{q^{h}} \\backslash\\mathbb{F}_{q}$, the set $S=\\{ \\alpha-x^t, x\\in\\mathbb{F}_{q}\\}$ of cardinality $1+\\frac {q-1} {M(h)}$ contains a non-d-th power i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.05976","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":""},"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":"1708.05976","created_at":"2026-05-18T00:36:02.660207+00:00"},{"alias_kind":"arxiv_version","alias_value":"1708.05976v2","created_at":"2026-05-18T00:36:02.660207+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1708.05976","created_at":"2026-05-18T00:36:02.660207+00:00"},{"alias_kind":"pith_short_12","alias_value":"PUKRKLYKG2DB","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_16","alias_value":"PUKRKLYKG2DBHLRO","created_at":"2026-05-18T12:31:37.085036+00:00"},{"alias_kind":"pith_short_8","alias_value":"PUKRKLYK","created_at":"2026-05-18T12:31:37.085036+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/PUKRKLYKG2DBHLRO5LIFNY2CFQ","json":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ.json","graph_json":"https://pith.science/api/pith-number/PUKRKLYKG2DBHLRO5LIFNY2CFQ/graph.json","events_json":"https://pith.science/api/pith-number/PUKRKLYKG2DBHLRO5LIFNY2CFQ/events.json","paper":"https://pith.science/paper/PUKRKLYK"},"agent_actions":{"view_html":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ","download_json":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ.json","view_paper":"https://pith.science/paper/PUKRKLYK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1708.05976&json=true","fetch_graph":"https://pith.science/api/pith-number/PUKRKLYKG2DBHLRO5LIFNY2CFQ/graph.json","fetch_events":"https://pith.science/api/pith-number/PUKRKLYKG2DBHLRO5LIFNY2CFQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ/action/storage_attestation","attest_author":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ/action/author_attestation","sign_citation":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ/action/citation_signature","submit_replication":"https://pith.science/pith/PUKRKLYKG2DBHLRO5LIFNY2CFQ/action/replication_record"}},"created_at":"2026-05-18T00:36:02.660207+00:00","updated_at":"2026-05-18T00:36:02.660207+00:00"}