{
  "id": "1709.04440",
  "version": "v1",
  "published": "2017-09-13T17:35:19.000Z",
  "updated": "2017-09-13T17:35:19.000Z",
  "title": "A polynomial bound for the arithmetic $k$-cycle removal lemma in vector spaces",
  "authors": [
    "Jacob Fox",
    "László Miklós Lovász",
    "Lisa Sauermann"
  ],
  "comment": "12 pages, including references",
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "For each $k\\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for any abelian group $G$. The best known bounds relating the parameters in the lemma for general $G$ are of tower-type. For $k>3$, even in the case $G=\\mathbb{F}_2^n$ no better bounds were known prior to this paper. This special case has received considerable attention due to its close connection to property testing of boolean functions. For every $k\\geq 3$, we prove a polynomial bound relating the parameters for $G=\\mathbb{F}_p^n$, where $p$ is any fixed prime. This extends the result for $k=3$ by the first two authors. Due to substantial issues with generalizing the proof of the $k=3$ case, a new strategy is developed in order to prove the result for $k>3$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-13T17:35:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05D99",
      "11B30"
    ],
    "keywords": [
      "cycle removal lemma",
      "polynomial bound",
      "vector spaces",
      "arithmetic",
      "abelian group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}