{
  "id": "2112.11752",
  "version": "v2",
  "published": "2021-12-22T09:37:15.000Z",
  "updated": "2022-06-29T07:49:24.000Z",
  "title": "Some Connections Between Discrepancy, Finite Gap Properties, and Pair Correlations",
  "authors": [
    "Christian Weiß"
  ],
  "comment": "to appear in: Monantshefte f\\\"ur Mathematik",
  "categories": [
    "math.NT",
    "math.CO"
  ],
  "abstract": "A generic uniformly distributed sequence $(x_n)_{n \\in \\mathbb{N}}$ in $[0,1)$ possesses Poissonian pair correlations (PPC). Vice versa, it has been proven that a sequence with PPC is uniformly distributed. Grepstad and Larcher gave an explicit upper bound for the discrepancy of a sequence given that it has PPC. As a first result, we generalize here their result to the case of $\\alpha$-pair correlations with $0 < \\alpha < 1$. Since the highest possible level of uniformity is achieved by low-discrepancy sequences it is tempting to assume that there are examples of such sequences which also have PPC. Although there are no such known examples, we prove that every low-discrepancy sequence has at least $\\alpha$-pair correlations for $0 < \\alpha <1$. According to Larcher and Stockinger, the reason why many known classes of low-discrepancy sequences fail to have PPC is their finite gap property. In this article, we furthermore show that the discrepancy of a sequence with the finite gap property plus a condition on the distribution of the different gap lengths can be estimated. As a concrete application of this estimation, we re-prove the fact that van der Corput and Kronecker sequences are low-discrepancy sequences. Consequently, it follows from the finite gap property that these sequences have $\\alpha$-pair correlations for $0 < \\alpha < 1$.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-06-29T07:49:24.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "connections",
      "finite gap property plus",
      "possesses poissonian pair correlations",
      "explicit upper bound",
      "low-discrepancy sequences fail"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}