{
  "id": "1907.03943",
  "version": "v1",
  "published": "2019-07-09T02:13:18.000Z",
  "updated": "2019-07-09T02:13:18.000Z",
  "title": "Congruences with intervals and arbitrary sets",
  "authors": [
    "William Banks",
    "Igor Shparlinski"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "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 of Kowalski, Michel and Sawin (2018).",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-09T02:13:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arbitrary set",
      "congruence",
      "trilinear character sums",
      "wide range",
      "kloosterman sums"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}