{
  "id": "1903.08266",
  "version": "v1",
  "published": "2019-03-19T21:43:33.000Z",
  "updated": "2019-03-19T21:43:33.000Z",
  "title": "Caps and progression-free sets in $\\mathbb{Z}_m^n$",
  "authors": [
    "Christian Elsholtz",
    "Péter Pál Pach"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "We study progression-free sets in the abelian groups $G=(\\mathbb{Z}_m^n,+)$. Let $r_k(\\mathbb{Z}_m^n)$ denote the maximal size of a set $S \\subset \\mathbb{Z}_m^n$ that does not contain a proper arithmetic progression of length $k$. We give lower bound constructions, which e.g. include that $r_3(\\mathbb{Z}_m^n) \\geq C_m \\frac{((m+2)/2)^n}{\\sqrt{n}}$, when $m$ is even. When $m=4$ this is of order at least $3^n/\\sqrt{n}\\gg \\vert G \\vert^{0.7924}$. Moreover, if the progression-free set $S\\subset \\mathbb{Z}_4^n$ satisfies a technical condition, which dominates the problem at least in low dimension, then $|S|\\leq 3^n$ holds. We present a number of new methods which cover lower bounds for several infinite families of parameters $m,k,n$, which includes for example: $r_6(\\mathbb{Z}_{125}^n) \\geq (85-o(1))^n$. For $r_3(\\mathbb{Z}_4^n)$ we determine the exact values, when $n \\leq 5$, e.g. $r_3(\\mathbb{Z}_4^5)=124$, and for $r_4(\\mathbb{Z}_4^n)$ we determine the exact values, when $n \\leq 4$, e.g. $r_4(\\mathbb{Z}_4^4)=128$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-03-19T21:43:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "exact values",
      "study progression-free sets",
      "cover lower bounds",
      "proper arithmetic progression",
      "lower bound constructions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}