{
  "id": "2311.01552",
  "version": "v1",
  "published": "2023-11-02T19:07:20.000Z",
  "updated": "2023-11-02T19:07:20.000Z",
  "title": "On the image of convolutions along an arithmetic progression",
  "authors": [
    "Ernie Croot",
    "Chi-Nuo Lee"
  ],
  "categories": [
    "math.CO"
  ],
  "abstract": "We consider the question of determining the structure of the set of all $d$-dimensional vectors of the form $N^{-1}(1_A*1_{-A}(x_1), ..., 1_A*1_{-A}(x_d))$ for $A \\subseteq \\{1,...,N\\}$, and also the set of all $(2N+1)^{-1}(1_B*1_B(x_1), ..., 1_B*1_B(x_d))$, for $B \\subseteq \\{-N, -N+1, ..., 0, 1, ..., N\\}$, where $x_1,...,x_d$ are fixed positive integers (we let $N \\to \\infty$). Using an elementary method related to the Birkhoff-von Neumann theorem on decompositions of doubly-stochastic matrices we show that both the above two sets of vectors roughly form polytopes; and of particular interest is the question of bounding the number of corner vertices, as well as understanding their structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-02T19:07:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "arithmetic progression",
      "convolutions",
      "vectors roughly form polytopes",
      "birkhoff-von neumann theorem",
      "corner vertices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}