{
  "id": "1606.02452",
  "version": "v1",
  "published": "2016-06-08T08:36:00.000Z",
  "updated": "2016-06-08T08:36:00.000Z",
  "title": "Packing near the tiling density and exponential bases for product domains",
  "authors": [
    "Mihail N. Kolountzakis"
  ],
  "categories": [
    "math.CA",
    "math.MG"
  ],
  "abstract": "A set $\\Omega$ in a locally compact abelian group is called spectral if $L^2(\\Omega)$ has an orthogonal basis of group characters. An important problem, connected with the so-called Spectral Set Conjecture (saying that $\\Omega$ is spectral if and only if a collection of translates of $\\Omega$ can partition the group), is the question of whether the spectrality of a product set $\\Omega = A \\times B$, in a product group, implies the spectrality of the factors $A$ and $B$. Recently Greenfeld and Lev proved that if $I$ is an interval and $\\Omega \\subseteq {\\mathbb R}^d$ then the spectrality of $I \\times \\Omega$ implies the spectrality of $\\Omega$. We give a different proof of this fact by first proving a result about packings of high density implying the existence of tilings by translates of a function. This allows us to improve the result to a wider collection of product sets than those dealt with by Greenfeld and Lev. For instance when $A$ is a union of two intervals in ${\\mathbb R}$ then we show that the spectrality of $A \\times \\Omega$ implies the spectrality of both $A$ and $\\Omega$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-06-08T08:36:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42C99",
      "52C22"
    ],
    "keywords": [
      "product domains",
      "exponential bases",
      "tiling density",
      "spectrality",
      "product set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}