{"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"}