{
  "id": "1909.09046",
  "version": "v1",
  "published": "2019-09-19T15:24:12.000Z",
  "updated": "2019-09-19T15:24:12.000Z",
  "title": "On the Wasserstein Distance between Classical Sequences and the Lebesgue Measure",
  "authors": [
    "Louis Brown",
    "Stefan Steinerberger"
  ],
  "categories": [
    "math.CA",
    "cs.NA",
    "math.FA",
    "math.NA",
    "math.NT"
  ],
  "abstract": "We discuss the classical problem of measuring the regularity of distribution of sets of $N$ points in $\\mathbb{T}^d$. A recent line of investigation is to study the cost ($=$ mass $\\times$ distance) necessary to move Dirac measures placed in these points to the uniform distribution. We show that Kronecker sequences satisfy optimal transport distance in $d \\geq 3$ dimensions. This shows that for differentiable $f: \\mathbb{T}^d \\rightarrow \\mathbb{R}$ and badly approximable vectors $\\alpha \\in \\mathbb{R}^d$, we have $$ \\ | \\int_{\\mathbb{T}^d} f(x) dx - \\frac{1}{N} \\sum_{k=1}^{N} f(k \\alpha) \\ | \\leq c_{\\alpha} \\frac{ \\| \\nabla f\\|^{(d-1)/d}_{L^{\\infty}}\\| \\nabla f\\|^{1/d}_{L^{2}} }{N^{1/d}}.$$ We note that the result is uniformly true for a sequence instead of a set. Simultaneously, it refines the classical integration error for Lipschitz functions, $\\| \\nabla f\\|_{L^{\\infty}} N^{-1/d}$. We obtain a similar improvement for numerical integration with respect to the regular grid. The main ingredient is an estimate involving Fourier coefficients of a measure; this allows for existing estimates to be conviently `recycled'. We present several open problems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-19T15:24:12.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lebesgue measure",
      "wasserstein distance",
      "classical sequences",
      "sequences satisfy optimal transport distance",
      "kronecker sequences satisfy optimal transport"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}