{
  "id": "1706.08023",
  "version": "v1",
  "published": "2017-06-25T02:03:35.000Z",
  "updated": "2017-06-25T02:03:35.000Z",
  "title": "On generalizations of $p$-sets and their applications",
  "authors": [
    "Heng Zhou",
    "Zhiqiang Xu"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT",
    "cs.CC",
    "cs.IT",
    "math.IT",
    "math.NA"
  ],
  "abstract": "The $p$-set, which is in a simple analytic form, is well distributed in unit cubes. The well-known Weil's exponential sum theorem presents an upper bound of the exponential sum over the $p$-set. Based on the result, one shows that the $p$-set performs well in numerical integration, in compressed sensing as well as in UQ. However, $p$-set is somewhat rigid since the cardinality of the $p$-set is a prime $p$ and the set only depends on the prime number $p$. The purpose of this paper is to present generalizations of $p$-sets, say $\\mathcal{P}_{d,p}^{{\\mathbf a},\\epsilon}$, which is more flexible. Particularly, when a prime number $p$ is given, we have many different choices of the new $p$-sets. Under the assumption that Goldbach conjecture holds, for any even number $m$, we present a point set, say ${\\mathcal L}_{p,q}$, with cardinality $m-1$ by combining two different new $p$-sets, which overcomes a major bottleneck of the $p$-set. We also present the upper bounds of the exponential sums over $\\mathcal{P}_{d,p}^{{\\mathbf a},\\epsilon}$ and ${\\mathcal L}_{p,q}$, which imply these sets have many potential applications.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-06-25T02:03:35.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11K38",
      "65C05"
    ],
    "keywords": [
      "generalizations",
      "well-known weils exponential sum theorem",
      "prime number",
      "upper bound",
      "goldbach conjecture holds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}