{
  "id": "2009.08846",
  "version": "v1",
  "published": "2020-09-18T14:05:31.000Z",
  "updated": "2020-09-18T14:05:31.000Z",
  "title": "Zero subsums in vector spaces over finite fields",
  "authors": [
    "Cosmin Pohoata",
    "Dmitriy Zakharov"
  ],
  "categories": [
    "math.CO",
    "math.NT"
  ],
  "abstract": "The Olson constant $\\mathcal{O}L(\\mathbb{F}_{p}^{d})$ represents the minimum positive integer $t$ with the property that every subset $A\\subset \\mathbb{F}_{p}^{d}$ of cardinality $t$ contains a nonempty subset with vanishing sum. The problem of estimating $\\mathcal{O}L(\\mathbb{F}_{p}^{d})$ is one of the oldest questions in additive combinatorics, with a long and interesting history even for the case $d=1$. In this paper, we prove that for any fixed $d \\geq 2$ and $\\epsilon > 0$, the Olson constant of $\\mathbb{F}_{p}^{d}$ satisfies the inequality $$\\mathcal{O}L(\\mathbb{F}_{p}^{d}) \\leq (d-1+\\epsilon)p$$ for all sufficiently large primes $p$. This settles a conjecture of Hoi Nguyen and Van Vu.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-18T14:05:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "vector spaces",
      "finite fields",
      "zero subsums",
      "olson constant",
      "minimum positive integer"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}