{
  "id": "2406.03275",
  "version": "v1",
  "published": "2024-06-05T13:53:05.000Z",
  "updated": "2024-06-05T13:53:05.000Z",
  "title": "Improved stability for the size and structure of sumsets",
  "authors": [
    "Andrew Granville",
    "Jack Smith",
    "Aled Walker"
  ],
  "comment": "15 pages",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "Let $A \\subset \\mathbb{Z}^d$ be a finite set. It is known that the sumset $NA$ has predictable size ($\\vert NA\\vert = P_A(N)$ for some $P_A(X) \\in \\mathbb{Q}[X]$) and structure (all of the lattice points in some finite cone other than all of the lattice points in a finite collection of exceptional subcones), once $N$ is larger than some threshold. In previous work, joint with Shakan, the first and third named authors established the first effective bounds for both of these thresholds for an arbitrary set $A$. In this article we substantially improve each of these bounds, coming much closer to the corresponding lower bounds known.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-06-05T13:53:05.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "lattice points",
      "finite set",
      "finite cone",
      "corresponding lower bounds",
      "finite collection"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}