{
  "id": "2408.15174",
  "version": "v1",
  "published": "2024-08-27T16:11:41.000Z",
  "updated": "2024-08-27T16:11:41.000Z",
  "title": "On Lev's periodicity conjecture",
  "authors": [
    "Christian Reiher"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We classify the sum-free subsets of ${\\mathbb F}_3^n$ whose density exceeds $\\frac16$. This yields a resolution of Vsevolod Lev's periodicity conjecture, which asserts that if a sum-free subset ${A\\subseteq {\\mathbb F}_3^n}$ is maximal with respect to inclusion and aperiodic (in the sense that there is no non-zero vector $v$ satisfying $A+v=A$), then $|A|\\le \\frac12(3^{n-1}+1)$ -- a bound known to be optimal if $n\\ne 2$, while for $n=2$ there are no such sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-08-27T16:11:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11B13",
      "11B30",
      "11P70"
    ],
    "keywords": [
      "sum-free subset",
      "vsevolod levs periodicity conjecture",
      "density exceeds",
      "non-zero vector",
      "resolution"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}