{
  "id": "1801.02164",
  "version": "v1",
  "published": "2018-01-07T09:32:57.000Z",
  "updated": "2018-01-07T09:32:57.000Z",
  "title": "Spectrality of product domains and Fuglede's conjecture for convex polytopes",
  "authors": [
    "Rachel Greenfeld",
    "Nir Lev"
  ],
  "categories": [
    "math.CA"
  ],
  "abstract": "A set $\\Omega \\subset \\mathbb{R}^d$ is said to be spectral if the space $L^2(\\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets \"behave like\" sets which can tile the space by translations. This suggests a conjecture that a product set $\\Omega = A \\times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope $\\Omega$ is spectral if and only if it can tile by translations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-07T09:32:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B10",
      "52C22"
    ],
    "keywords": [
      "convex polytope",
      "product domains",
      "fugledes conjecture",
      "spectral sets",
      "spectrality"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}