{"paper":{"title":"Density of non-residues in Burgess-type intervals and applications","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"D. R. Heath-Brown, I. E. Shparlinski, M. Z. Garaev, W. D. Banks","submitted_at":"2006-07-27T00:13:43Z","abstract_excerpt":"We show that for any fixed $\\eps>0$, there are numbers $\\delta>0$ and $p_0\\ge 2$ with the following property: for every prime $p\\ge p_0$ and every integer $N$ such that $p^{1/(4\\sqrt{e})+\\eps}\\le N\\le p$, the sequence $1,2,...,N$ contains at least $\\delta N$ quadratic non-residues modulo $p$. We use this result to obtain strong upper bounds on the sizes of the least quadratic non-residues in Beatty and Piatetski--Shapiro sequences."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0607692","kind":"arxiv","version":3},"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"}