{"paper":{"title":"Congruences with intervals and arbitrary sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Igor Shparlinski, William Banks","submitted_at":"2019-07-09T02:13:18Z","abstract_excerpt":"Given a prime $p$, an integer $H\\in[1,p)$, and an arbitrary set $\\cal M\\subseteq \\mathbb F_p^*$, where $\\mathbb F_p$ is the finite field with $p$ elements, let $J(H,\\cal M)$ denote the number of solutions to the congruence $$ xm\\equiv yn\\bmod p $$ for which $x,y\\in[1,H]$ and $m,n\\in\\cal M$. In this paper, we bound $J(H,\\cal M)$ in terms of $p$, $H$ and the cardinality of $\\cal M$. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1907.03943","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"}